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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2009/
Module Title: Engineering Mathematics 311
Module Code: MATH
School: School of Building & Civil
Programme: Bachelor of Engineering (Honours) in Structural Engineering
Programme Code: CSTRU_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Answer 4 questions. These questions carry equal marks.
Duration: 2 Hours
Sitting: Autumn 2010
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
Q1. (a) Show that the matrices below have the same eigenvalues but different eigenvectors
A =
^
B =
Deduce whether or not that the eigenvectors of A are linearly independent and mutually orthogonal. (16marks)
(b) In a vibrating system the displacements of two masses from their equilibrium positions are found by solving the system of differential equations:
22 1 2
2
21 1 2
2
4 x 12x dt
d x
d x
By assuming solutions of the form xi=Ricos(t-i) find the general solution of the set of simultaneous differential equations above. (9 marks)
Q2. (a) A function f(x)=L-x is defined over the interval [0,L]. Plot an odd extension of f(x)
and find the corresponding Half Range Fourier Series.
Note:
L
n x n
L
L
n x n
LL x L
L x n x^
( )cos ^ ( )sin cos
2 2
2
L
n x n
L
L
n x n
LL x L
L x n x^
( )sin ^ ( )cos sin 2 2
2 (10 marks)
(b) The temperature u(x,t) at any point on the rod aligned along the x-axis between the points x=0 and x=L at any instant t is found by solving the partial differential equation
2 2
u (^) k u t x
Both ends are maintained at 20^0 C. The initial temperature distribution is given by u(x,0)=g(x). By using a substitution v(x,t)=u(x,t)-20 solve this partial differential equation. In particular find the solution when g(x)=50. (15 marks)
Q4. (a) Find the Inverse Laplace Transform of the expressions
(i) 2 2 (s +4)
64s (ii) .... s (s +4)
16e s (s +4)
8e 2 2
-3s 2 2
-s (^) (14 marks)
(b) By using Laplace transforms solve the differential equation
2x 10 U(t 2) x(0) x(0) 0 dt
3 dx dt
d x 2
2 (7 marks)
(c) The deflection y at any point of a beam is found by solving the differential equation
2 dx
EId y -20(x-3)+R(x-4)
where R is a constant. By using Laplace transforms solve this differential equation where the end x=0 is fixed. At x=5 the deflection y is zero at x=0. (5 marks)
Q5. (a) A light beam is of span 4m has both ends embedded in walls. Between the points
x=0m and x=2m there is a U.D.L. of 72kNm-1. Express w the load per unit length in terms of the Heaviside Unit Step Function. By using Laplace Transforms solve the differential equation 4 4 EI d y w dx
to find the deflection y at any point on the beam. Note: L[f iv^ (t)]s^4 F(s)s^3 f(0)s^2 f(0)sf(0)f///(0) (10 marks)
(b) Find the eigenvalues and the corresponding eigenvectors of the matrix
A =
Does an ortononal matrix P exist where PTAP is diagonal? Justify your answer. If it exists find the matrix P and verify that it is orthogonal. (1 0 marks)
(c) By using spherical coordinates (r,find the volume of the sphere described by x^2 +y^2 +z^2 ≤ The Jacobian is given by J=r^2 sin. (5 marks)
DERATIVES
f(x) f (x) a=constant sinx cosx cosx -sinx eax^ a eax uv u dv (^) +vdu dx dx
INTEGRALS
f(x) f(x)dx a=constant
sinx - cosx cosx sinx
sin^2 A=^1 2
(1-cos2A) cos^2 A=^1 2
(1-cos2A)
sin(-A)=-sinA cos(-A)=cosA
2sinAcosB=sin(A+B)+sin(A-B) 2cosAcosB=cos(A+B)+cos(A-B)
2sinAsinB=cos(A-B)-cos(A+B)
A 0 2
sinA 0 0 0 cosA 1 - 1 1
