Download Mechanical System - Engineering Mathematics - Exam and more Exams Engineering Mathematics in PDF only on Docsity!
CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2009/
Module Title: 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-Stage 2 Bachelor of Engineering (Honours) in Biomedical Engineering-Stage 2
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 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.
- (a) Find the Inverse Laplace Transform of the expressions
(i) (^2 ) (s 4 )
(ii) s (s 4 )
8e 2 2
-3s
(14 marks)
(b) In a mechanical system the response y(t) to an input f(t) is found by solving the differential equation
ky f(t) y(0) y(0) 0 dt
cdy dt
m d y 2
2
By using Laplace transforms solve this differential equation where (i) m=1, c=4, k=3, f(t)=10δ(t-4), (ii) m=1, c=0, k=0 and f(t) is the function defined by
f(t)=
8 - 2t ift 2
4 if 0 t (^2) (11 marks)
- (a) Find the work done when the force F =6xy i +12y 2 j moves a mass along the perimeter
of the triangular region with vertices (-2,0), (2,0) and (0,2). (10 marks)
(b) A volume V has a constant cross sectional area and this volume is decribed by
V: x^2 +y^2 ≤ 4 0 ≤z≤3.
(i) If C is the perimeter of the base evaluate the line integral
∫ C +
2x^2 dx 6xydy
(ii) Evaluate the surace integral below over the base and over the top of this volume
∫∫ S + +
(3x^2 3y^2 3z^2 )dA
(iii) For this volume evaluate the triple integral
∫∫∫V
4y 2 z^2 dV (15 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 in R then
C R
)dA y
f(x, y)dx+g(x,y)dy ( g.
where the direction of C is anticlockwise. Verify Green’s Theorem where f(x,y)=30x 2 +30y^2 , g(x,y)=120xy and where R is the region in the first quadrant bounded by the line y=x and the parabola y=x^2. (11 marks)
(b) Find the eigenvalues and the corresponding eigenvectors of the matrix
A=
(i) Show that the eigenvalues of this matrix are linearly independent and mutually orthogonal. Find an orthogonal matrix P where PT^ AP is diagonal. Verify that P is orthogonal.
(ii) By assuming exponential solutions find the general solution of the system of differential equations
d (^) = dt
x (^) Ax (14 marks)
Z-TRANSFORMS
f(t) F(z) U(n)= z 1
z − a^ N z a
z − n (z 1)^2
z − n 2 (z 1)^3
z(z 1) −
a n f(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) x n nx n−^1 e ax a eax sinx cosx cosx -sinx
f(x) a=constant ∫ f(x)dx
x n xn+ if n - n+
e ax^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)
