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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2008/
Module Title: Engineering Mathematics 202
Module Code: MATH
School: Mechanical & Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2
Programme Code: EMECH_8_Y
External Examiner(s): Dr.P. Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Select any four questions. The questions carry equal marks.
Duration: 2 Hours
Sitting: Autumn 2009
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.
- (a) Find the eigenvalues and the corresponding eigenvectors of the matrices
A B =
Show that the eigenvectors of A are linearly independent but are not mutually orthogonal. (16 marks)
(b) Two masses are attached to two springs. The displacements of these masses from their equilibrium positions are found by solving the set of simultaneous differential equations
1 1 2 2 1 2
x 5x 2x x 2x 2x
By assuming periodic solutions of the form xi =Ri cos(ωt-αi ) find the general solution of this set of differential equations. (9 marks)
- (a) Find the Inverse Laplace Transform of the expressions below.
(i) 2 16s 2 (s +4)
(ii)
-3s 3
8e s
(9 marks)
(b) By using Laplace Transform solve the differential equations
(i)
(^2) -t 2
d y (^) +3 dy+2y=12e y (0)=y(0)= dt dt
(ii)
2 2
d y (^) 4y 16U(t-1) y (0) y(0) 0 dt
4 (a) (i) Find the first three sampled values of the function whose z Transform is given by
the expression
2
2z z -4z+
Use long division and use partial fractions.
(ii) With the aid of the tables of z-Transforms solve the difference equation
yn+2 -3yn+1 +2yn =8(2) n^ y 0 =y 1 =0 (15 marks)
(b) Express two cycles of the triangular wave below in terms of the Heaviside Unit Step Function and find its Laplace Transform.
(10 marks)
t
f(t)
(1,1) (2,1)^ (3,1)
- (a) By summing vertically and by summing horizontally find the area of the region
in the first quadrant bounded by the line y=x and the parabola y=x^2. (9 marks)
(b) Find the eigenvalues and the corresponding eigenvectors of the matrix
A =
æç - ö÷ çç ÷÷ çç ÷÷÷ ççè (^) ÷÷ø
Hence find the general solution of the general solution of the system of differential equations
d (^) = dt
x (^) Ax
(11 marks)
(c) Find the Inverse Laplace Transform of the expression below and sketch the function Obtained
-s -2s 2 2 2
(^2) - 4e 4e s s s
LAPLACE TRANSFORMS
For a function f(t) the Laplace Transform of f(t) is a function in s defined by
∞ = − 0
F(s) est^ f(t)dt where s>0.
f(t) F(s) A=constant A s t n sn^1
n!
e at^1 s −a sinhkt (^) k s 2 −k^2 coshkt s s 2 −k^2
sin ωt ω
s 2 + ω^2
cos ωt s
s 2 +ω^2
e at^ f(t) F(s-a) f (t) ′ sF(s)-f(0) f (t) ′′ (^) s F(s)^2 − sf(0) − f (o)′
f(u)du 0
t
F(s) s
f(u)g(t u)du 0
t
F(s)G(s)
U(t-a) (^) e s
-as
f(t-a)U(t-a) (^) e −asF(s) δ ( t − a) e -as
Note: coshA
e e 2 sinhA^
e e 2
A A A A
