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CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2010/
Module Title: MATH7005: Engineering Mathematics 202
Module Code: MATH
School: School of Mechanical & Process Engineering School of Manufacturing, Biomedical & Facilities Engineering
Programme Title:Bachelor of Engineering (Honours) in Mechanical Engineering Bachelor of Engineering (Honours) in Biomedical Engineering
Programme Code: EMECH-8-Y EBIOM-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: Summer 2011
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.
- The displacement x of a mass attached to a spring and a dashpot at any instant t
is found by solving the differential equation
kx f(t) x(0) x(0) 0 dt
cdx dt
m dx 2
2
By using Laplace Transforms solve this differential equation in the cases where (i) m=1, c=4, k=4, f(t)=60(t-1), (3 marks) (ii) m=1, c=0, k=9and f(t)=36cos3t, (6 marks) (iii) m=1, c=0, k=9 and f(t)=36(t-1)U(t-1)-72(t-2)U(t-2) (9 marks) (iv) m=1, c=0, k=0 and f(t) is defined by
f(t)=
6 - 3t ift 2
3t if 0 t 2 (7 marks)
- (a) Green’s Theorem states: If C is a piecewise smooth closed curve that encloses a region
R and if f(x,y) and g(x,y) have continuous partial derivatives throughout R then
C R
)dA y
f(x,y)dx+g(x,y)dy ( g
where the direction of C is anticlockwise. Verify Greens Theorem where f(x,y)=6y^2 , g(x,y)=18xy and R is the triangular region with vertices (-1,0), (1,0) and (1,4). ( 12 marks)
(b) The region R is the elliptical region defined by
y 4
x (^2) ^2
(i) If C is the perimeter of this region evaluate the line integral
C
3xydx 12y^2 dy
(ii) By evaluating an appropriate double integral find the second moment of area of this region about the y-axis. (13 marks)
(b) The displacements of three masses from their equilibrium positions are found by solving the system of differential equations
3 1 2 3
2 1 2 3
1 1 2 3
x 21x 3x 21x
x 12x 9x 6x
x 1 2x 3x 12x
(9 marks)
By assuming solutions of the from xi = Ri Cos (ωt – αi) solve this set of equations.
5 (a) By using partial fractions and by using long division find the first four sampled values
whose z-transform is given by
(z -4z 5z-2)
4z (z-2)(z-1)
4z 3 2
2 2
2
(10 marks)
(b) Use z-Transforms to solve the difference equations
yn+2-4yn+1+4yn=0 y 0 =6 y 1 =0 (7 marks)
(c) Find the Laplace Transform of two cycles of the sawtooth wave below
…
(8 marks)
t
f(t)
(0,0) (1,0)^ (2,0)
(1,2) (^) (2,2)
Z-TRANSFORMS
f(t) F(z) U(n)= z 1
z aN z a
z n (z 1)^2
z n^2 (z 1)^3
z(z 1)
anf(n)
a
F z nf(n) -zF(z)
f(n+1) zF(z)-zf(0)
f(n+2) z 2 F(z)^ z^2 f(0)zf(1)
DERIVATIVES AND INTEGRALS
f(x) a=constant f (x) xn nxn ^1 eax a eax sinx cosx cosx - sinx
f(x) a=constant f(x)dx
xn xn+ if n - n+
eax^1 a
a eax sinx - cosx cosx sinx
Note : 2sinAcosB=sin(A+B)+sin(A-B) 2cosAcosB=cos(A+B)+cos(A-B)
2sinAsinB=cos(A-B)-cos(A+B) sin(-A)=-sinA cos(-A)=cosA
cos^2 A= 2
1 (1+cos2A) sin^2 A= 2
1 (1-cos2A)
