Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version
8th Edition
ISBN: 9781119080701
Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein
Publisher: WILEY
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Chapter 3.6, Problem 105P
To determine
Estimate the amount of time it will take for the boat to sink.
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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 3 Solutions
Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version
Ch. 3.2 - Prob. 1PCh. 3.2 - Air flows steadily along a streamline from point...Ch. 3.2 - Water flows steadily through the variable area...Ch. 3.2 - What pressure gradient along the streamline,...Ch. 3.2 - At a given location the airspeed is 20 m/s and the...Ch. 3.2 - What pressure gradient along the streamline,...Ch. 3.2 - The Bernoulli equation is valid for steady,...Ch. 3.2 - An incompressible fluid flows steadily past a...Ch. 3.2 - Consider a compressible liquid that has a constant...Ch. 3.3 - Air flows along a horizontal, curved streamline...
Ch. 3.3 - Water flows around the vertical two-dimensional...Ch. 3.3 - Water in a container and air in a tornado flow in...Ch. 3.3 - Prob. 15PCh. 3.5 - At a given point on a horizontal streamline in...Ch. 3.5 - A drop of water in a zero-g environment (as in the...Ch. 3.5 - When an airplane is flying 200 mph at 5000-ft...Ch. 3.5 - Air flows over the airfoil shown in Fig. P3.20....Ch. 3.5 - Some animals have learned to take advantage of the...Ch. 3.5 - Estimate the pressure on your hand when you hold...Ch. 3.5 - 2013 Indianapolis 500 champion Tony Kanaan holds...Ch. 3.5 - What is the minimum height for an oil (SG = 0.75)...Ch. 3.5 - Prob. 25PCh. 3.5 - A Bourdon-type pressure gage is used to measure...Ch. 3.5 - Estimate the force of a hurricane strength wind...Ch. 3.5 - A 40-mph wind blowing past your house speeds up as...Ch. 3.5 - Prob. 29PCh. 3.6 - Prob. 30PCh. 3.6 - Estimate the pressure needed at the pumper truck...Ch. 3.6 - The tank shown in Fig. P3.32 contains air at...Ch. 3.6 - Water flows from the faucet on the first floor of...Ch. 3.6 - Prob. 34PCh. 3.6 - Prob. 35PCh. 3.6 - Streams of water from two tanks impinge upon each...Ch. 3.6 - Several holes are punched into a tin can as shown...Ch. 3.6 - Water flows from a pressurized tank, through a...Ch. 3.6 - Prob. 39PCh. 3.6 - Prob. 41PCh. 3.6 - Figure P3.42 shows a tube for siphoning water from...Ch. 3.6 - For the pipe enlargement shown in Fig. P3.43, the...Ch. 3.6 - A fire hose nozzle has a diameter of in. According...Ch. 3.6 - Water flowing from the 0.75-in.-diameter outlet...Ch. 3.6 - Prob. 46PCh. 3.6 - Prob. 47PCh. 3.6 - Prob. 48PCh. 3.6 - The pressure and average velocity at point A in...Ch. 3.6 - Water (assumed inviscid and incompressible) flows...Ch. 3.6 - Prob. 51PCh. 3.6 - Prob. 52PCh. 3.6 - Prob. 53PCh. 3.6 - Prob. 54PCh. 3.6 - Prob. 55PCh. 3.6 - Prob. 56PCh. 3.6 - Water (assumed frictionless and incompressible)...Ch. 3.6 - Prob. 58PCh. 3.6 - Water flows through the pipe contraction shown in...Ch. 3.6 - Prob. 60PCh. 3.6 - Prob. 61PCh. 3.6 - Prob. 62PCh. 3.6 - Prob. 63PCh. 3.6 - Prob. 64PCh. 3.6 - The circular stream of water from a faucet is...Ch. 3.6 - Water is siphoned from the tank shown in Fig....Ch. 3.6 - Prob. 67PCh. 3.6 - Prob. 68PCh. 3.6 - Water is siphoned from the tank shown in Fig....Ch. 3.6 - Prob. 70PCh. 3.6 - Water exits a pipe as a free jet and flows to a...Ch. 3.6 - Water flows steadily from a large, closed tank as...Ch. 3.6 - Prob. 73PCh. 3.6 - Prob. 74PCh. 3.6 - Prob. 75PCh. 3.6 - Prob. 76PCh. 3.6 - Prob. 77PCh. 3.6 - Prob. 78PCh. 3.6 - Prob. 79PCh. 3.6 - Air is drawn into a small open-circuit wing tunnel...Ch. 3.6 - Prob. 81PCh. 3.6 - Water flows steadily from the large open tank...Ch. 3.6 - Prob. 83PCh. 3.6 - Prob. 84PCh. 3.6 - Prob. 85PCh. 3.6 - Prob. 86PCh. 3.6 - Prob. 87PCh. 3.6 - Prob. 88PCh. 3.6 - Prob. 89PCh. 3.6 - Prob. 90PCh. 3.6 - Prob. 91PCh. 3.6 - Prob. 92PCh. 3.6 - Prob. 93PCh. 3.6 - Prob. 94PCh. 3.6 - Prob. 95PCh. 3.6 - Prob. 96PCh. 3.6 - Prob. 97PCh. 3.6 - Prob. 98PCh. 3.6 - Prob. 99PCh. 3.6 - Determine the flowrate through the submerged...Ch. 3.6 - The water clock (clepsydra) shown in Fig. P3.101...Ch. 3.6 - Prob. 102PCh. 3.6 - Prob. 105PCh. 3.6 - Prob. 106PCh. 3.6 - Prob. 107PCh. 3.6 - Prob. 109PCh. 3.6 - Prob. 110PCh. 3.6 - Water flows through the branching pipe shown in...Ch. 3.6 - Prob. 112PCh. 3.6 - Prob. 113PCh. 3.6 - Prob. 114PCh. 3.6 - Prob. 115PCh. 3.6 - Prob. 116PCh. 3.6 - Prob. 117PCh. 3.6 - Prob. 118PCh. 3.6 - Prob. 119PCh. 3.6 - Prob. 120PCh. 3.6 - Prob. 121PCh. 3.6 - Prob. 122PCh. 3.6 - Prob. 123PCh. 3.6 - Water flows in a rectangular channel that is 2.0 m...Ch. 3.6 - Prob. 125PCh. 3.6 - A Venturi meter with a minimum diameter of 3 in....Ch. 3.6 - Prob. 127PCh. 3.6 - Prob. 128PCh. 3.6 - What diameter orifice hole, d, is needed if under...Ch. 3.6 - A weir (see Video V10.13) of trapezoidal cross...Ch. 3.6 - Prob. 131PCh. 3.6 - Water flows under the inclined sluice gate shown...Ch. 3.7 - Water flows in a vertical pipe of 0.15-m diameter...Ch. 3.7 - Prob. 134PCh. 3.7 - Draw the energy line and hydraulic grade line for...Ch. 3.8 - Prob. 137PCh. 3.8 - Prob. 138P
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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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