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 (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.рv(x)XLThe PDE describing mechanical equilibrium for the string isd dvdxdx- 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 de-flection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.

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Answered: Problem 1 (65 pts, suggested time 50…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1
T 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 amount
v(x) from its undeformed (horizontal) state, as shown in the figure below.
The PDE describing mechanical equilibrium for the string is
d
dx
T
dv
dx
− p = 0 . (1)
(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Write
down 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 deflection
of 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 each
element.