PH.D. PRELIMINARY EXAMINATION MATHEMATICS


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1 UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering FALL SEMESTER 2014 Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem 1: Consider the following linear system of ordinary differential equations, dy 1 dt dy 2 dt dy 3 dt = y 2 (t) + y 3 (t) (1) = y 1 (t) + y 3 (t) (2) = y 1 (t) + y 2 (t). (3) Find the general solution for the following ordinary differential equations. (50 points) Problem 2: Show that the Laplace equation in twodimensional space, can be converted into a form of polar coordinates as where 1 r 2 Φ = 2 Φ x Φ = 0, (4) y2 ( r r Φ r ) + 1 r 2 2 Φ θ 2 = 0. x = r cos θ, y = r sin θ; or r = x 2 + y 2, tan θ = y x. If the problem is axisymmetric, i.e. Φ = Φ(r), what is the basic form of the solution for the equation. That is the solution satisfying (50 points) 1 r d ( dr r dφ dr ) = 0. 1
2 UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem 1 (50 points) Consider the following 3 3 matrix, (1) Find all the eigenvalues of the matrix; (2) Find the corresponding eigenvectors Problem 2 (50 points) Consider the following equilibrium equation of an elastic bar, with boundary conditions, Assume that ɛ << 1 and d dx (E(y)A)duɛ + b(x) = 0, 0 < x < L (1) dx u ɛ (0) = 0, and P (L) = E(y)A duɛ dx d dx = x + 1 ɛ y x=l = P. u ɛ (x) = u 0 (x, y) + ɛu 1 (x, y) + ɛ 2 u 2 (x, y) + (3) Substitute Eqs. (2) and (3) into (1), and expand the equilibrium equation of the elastic bar into a series expansion of ɛ. Derive the first three equations that corresponding to the lowest three exponents (scales) of ɛ, i.e. ɛ 2, ɛ 1, and ɛ 0. (2) 1
3 UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering FALL SEMESTER 2013 Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem 1 (50 points) Consider the following linear equation, (x 1) 0 x x 0 (x 1) y 1 y 2 y 3 = x 2 5 x 2 x + 1/4 (1) Find all the eigenvalues of the coefficient matrix; (2) Find all corresponding eigenvectors; (3) Discuss the solution of the linear algebraic equation {y i } when x 1/2; Problem 2 (50 points) Consider the following EulerBernoulli beam equation, with the boundary conditions, d 2 dx 2 EI(x) d2 v + kv(x) = 0, dx2 0 < x < L v(0) = 0, v (0) = 0; V (L) := d dx EI(x) d2 v dx 2 x=l = P L, M(L) := EI(x) d2 v dx 2 x=l = 0. Choose an arbitrary continuous function w(x) H 2 ([0, L]) and w(0) = 0, dw dx x=0 = 0. (1) Use the weak form of the differential equation to solve w(l) in terms of definite integrations of v(x) and w(x) and their second derivatives. Note that E, I(x) and k are either given constants or known function; (2) Choose w(x) = v(x) and solve v(l) in terms of E, P L, L and I 0 by assuming that I(x) = I 0 (L x) and k = 0. Hint: First observe and verify that M(x) := EI(x) d2 v dx 2 = P L(L x). 1
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6 University of California Department of Civil and Environmental Engineering Spring 2012 Doctoral Preliminary Exam: Mathematics Name: Problem #1 A function y(t) is known to satisfy the differential equation ÿ 2ẏ +5y =10sin(t) where ẏ and ÿ denote the first and second derivatives, respectively, of the function y(t) with respect to the independent variable t. Ify(0) = 1 and ẏ(0) = 0, find the value of y(t) and ẏ(t) fort = π. Problem #2 Let the 2 2matrixA be given by A = [ ] Find all matrices X that satisfy the equation How many of such X s are there? X 2 3X = A
7 UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering SPRING SEMESTER 2011 Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem 1: Consider the following definite integral, I[y] = 1 2 l 0 ( ( dy ) ) 2 + y 2 dx <, dx where y = y(x) is a continuous real function defined on the domain [0, l]. Obviously, the value of I depends on the function y(x), i.e. I = I[(y(x)]. Consider a real parameter α > 0 and another real function η(x), x [0, l] and η(0) = η(l) = 0. (1) Write the explicit expression of I[y(x) + αη(x)]; (2) Calculate the following limit, (3) Show that Hint: Use integration by parts. I(y + αη) I(y) δi := lim ; α 0 α δi = l 0 ( d 2 y dx 2 y) η(x)dx. Problem 2. Find the general solution for the following ordinary differential equation, d 2 y dt 2 + 2ζω dy 0 dt + ω2 0y = 0, with y(0) = 0, ẏ(0) = 1 ; Discuss your solution for the following cases: (1) ζ > 1, (2) ζ = 1, (3) 0 < ζ < 1, and (4) ζ = 0. 1
8 UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering FALL SEMESTER 2010 Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem 1: Find the general solution for the following ordinary differential equation, d 2 f dx df x dx 22 x 2 f = 0 (1) Problem 2: For the following matrix, [A] = (1) Find all the eigenvalues; (2) Find all the eigenvectors; (3) Find a transformation matrix [Q] such that is a diagonal matrix [B] = [Q] T [A][Q] (2) 1
9 Mathematics Preliminary Exam Spring 2010 Structural Engineering, Mechanics, and Materials 1. Consider the following PDE u x + u t = 0, defined for u(x, t) over x (, ) and t [0, ). (a) (15 pts) Show that every function u(x, t) = f(x t) qualifies as a solution to this equation, independent of the functional form for f( ). (b) (15 pts) Assume that the initial condition is given as u(x, 0) = H(x + 1) H(x 1), where H( ) is the Heaviside step function; i.e. u(x, 0) is a square pulse centered about the origin with a width of 2 units. Accurately sketch the solution at time 2; i.e. sketch u(x, 2). 2. Consider the quadratic form q = x T Ax. (a) (15 pts) Prove that q is a real number, if A is Hermitian 1 (independent of x). (b) (15 pts) Prove that q is a pure imaginary number (or zero), if A is skewhermitian 2 (independent of x). 3. (40 pts) Consider a region of space Ω = {(x, y, z) x 2 +y 2 < 25 and x+z < 10 and z +y > 4} with piecewise smooth boundary Ω. Using the divergence theorem, compute the integral I = n(x)ds, where n is the outward unit normal vector field on Ω. Ω 1 Hermitian: A = A, i.e. A ij = Āji. 2 skewhermitian: A = A, i.e. A ij = Āji. 1
10 Mathematics Preliminary Exam Fall 2009 Structural Engineering, Mechanics, and Materials 1. Consider the linear system of equations Ax = b, where 1 1 A = (a) (10pts) What condition must b satisfy for this system to have a solution? (b) (10pts) Assume that Find x. b = (c) (5pts) Outline how you would find a solution to the system of equations if b did not satisfy the condition you have outlined in Part 1a. 2. Consider the ordinary differential equation where x(0) = 1 and dx dt (0) = 0. (a) (20pts) Find x(t) for t > d 2 x dt 2 + x = sin(t), (b) (5pts) Qualitatively, sketch your solution. 3. (25pts) Assume φ(x, y) satisfies Laplace s equations ( 2 φ = 0) over the domain [0, 1] [0, 1] and is subject to the boundary conditions φ(0, y) = φ(x, 0) = φ(x, 1) = 0 and φ x (1, y) = sin(πy). Find φ(x, y) using separation of variables. 1
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14 University of California at Berkeley Spring 2008 Department of Civil and Environmental Engineering SEMM Doctoral Preliminary Examination: Mathematics Name Problem 1. Show that (20 points) x 1 x 2 x 3 x 2 1 x 2 2 x 2 3 = (x 2 x 1 )(x 3 x 1 )(x 3 x 2 ) (1) Problem 2. Consider the following differential equation, EI d4 y dx 4 = w 0H(x a), 0 < a, x < L (2) where w 0 is a constant, and H(x a) is the Heaviside function defined as H(x a) = { 1, x > a 0, x a (3) The boundary conditions of the differential equation are y(0) = y(l) = 0 (4) y (0) = y (L) = 0 (5) Find y(x). (35 points) Problem 3. Consider a twodimensional Laplace equation, in Cartesian coordinate. Use polar coordinates r and θ 2 f x f y 2 = 0 (6) x = r cos θ, r 2 = x 2 + y 2 (7) y = r sin θ, tan θ = y x (8)
15 and chain rule to show that (1) x = cos θ r sin θ r (2) θ and y = sin θ r + cos θ r θ (9) 2 f x f ( 2 y 2 = r r r ) r 2 θ 2 f = 0 (10) (3) If f is independent of r, find the general solution of f (in terms of θ and arbitrary constants); and (4) If f is independent of θ, find the general solution of f (in terms of r and arbitrary constants). ) (Hint: (45 points) 2 f dr df r dr = 1 d (r df ) r dr dr
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