Flange Widths - Mathematics - Exam, Exams of Mathematics

Download Flange Widths - Mathematics - Exam and more Exams Mathematics in PDF only on Docsity! Cork Institute of Technology Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 3 Bachelor of Engineering in Mechanical Engineering-Stage 3 (Level 8) Autumn 2005 Mathematics and Statistics (Time: 3½ Hours) Answer FIVE questions, at least TWO questions from each Section. Use separate answer books for each Section. All questions carry equal marks. Statistical tables are available. Examiners: Mr. J. Hegarty Prof. J. Monaghan Mr. D. O’Hare Mr.T.O Leary Section A 1. (a) Certain shipments of insulators were subject to inspection by a high - voltage laboratory. One of the important tests was a destructive one. The procedure adopted was as follows: Select 6 insulators at random from the batch and test them. If all 6 pass the test, accept the batch. If 2 or more fail, reject the batch. If only 1 insulator fails, take a second sample of 6 insulators. If all 6 insulators in the second sample pass the test, accept the batch; otherwise reject it. What is the probability of accepting batches in which 15% of insulators are non- conforming? (5 marks) (b) A discrete random variable X is distributed as follows: r 100 200 300 400 P(X=r) 0.25 k 0.1 0.1 Find the value of k and calculate the expected value and the variance of X. (5 marks) (c) (i) If the probability that a single article drawn from a continuous manufacturing process does not meet specifications is 0.08, what is the probability that a sample of 50 articles drawn from that process will contain 4 nonconforming items ? (ii) Using the Poisson distribution as an approximation, answer the question in part (i). (5 marks) 2 (d) Serious paint blemishes on a flat automotive panel occur at a rate of one blemish for every ten body panels with blemishes occurring according to the pattern of a Poisson distribution. (i) What is the probability of more than one serious blemish on a randomly chosen body panel? (ii) How many body panels should be inspected in order to have a probability greater than 0.99 of observing at least one serious blemish? (5 marks) 2. (a) Experience has shown that the width, in mm, of the flange on a plastic connector has the following distribution:    ≤≤ = otherwise,0 52.048.0for ,50 )( xx xf (i) Verify that this is a well-defined probability density function, and sketch its graph. (ii) Of the next 10000 connectors produced, how many do you estimate will have flange widths between 0.50 and 0.51mm? (7 marks) (b) (i) An exponentially distributed random variable, X, has probability density function 0,)( >= − xexf xλλ , and moment generating function .)( t tM X − = λ λ Show, using the moment generating function or otherwise, that the mean and the standard deviation of X are both equal to .1 λ (ii) The waiting time, in minutes, before receiving attention at a customer service desk is exponentially distributed. If the proportion of all customers who wait more than 10 minutes is 0.01, what is the mean waiting time for all customers? (8 marks) (c) Three parts are assembled in series so that their critical dimensions x1, x2 and x3 add. The dimension of each part is normally distributed and the following are the relevant parameters: .2,75,4,75,4,100 332211 ====== σµσµσµ What is the probability that an assembly chosen at random will have a combined dimension in excess of 262? (5 marks) 5 Section B 5. (a) Find the Laplace Transform of two cycles of the sawtooth wave defined by f(t)=12t where 1t0 ≤≤ f(t+1)=f(t). (4 marks) (b) In a mechanical system the response y(t) due to an input f(t) is found by solving the differential equation 0y(0)(0)yf(t)ky dt dyc dt ydm 2 2 ==′=++ . By using Laplace Transforms solve this differential equation where (i) m=1, c=4, k=4, f(t)=4e-2t, (ii) m=1, c=0, k=4, f(t)=16cos2t, (iii) m=1, c=3, k=2 and f(t) is two cycles of the wave in part (a). (16 marks) 6 6. (a) Find the eigenvalues and the corresponding eigenvectors of the matrices           = 422 224 242 A           = 600 224 242 B (i) Does an orthogonal matrix P exist where PTAP is diagonal? Justify your answer. If it exists find the matrix P and write down the matrix PTAP. You are not required to complete the diagonalisation (ii) Find the general solution of the set of simultaneous differential equations represented in the matrix form Axx = dt d (14 marks) (b) The displacement of three masses attached to three springs are found by solving the set of simultaneous differential equations 3213 3212 3211 36x32xx32x 5x2x14xx 5x2x18xx −−=″ +−−=″ ++−=″ By assuming periodic solutions of the form xi=Risin(ωt-αi) find the three values of ω and any one of the set of solutions. (6 marks) 7 7. (a) Find the Fourier Series for the function defined by f(t)=    ≤≤ ≤≤− 0tπ- ift - π- πt0 ift π f(t+2π)=f(t) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ − − −=− − − =− 2 2 n nxsinnxcos n x)(πdxnxx)sin(π n nxcosnxsin n x)(πdxnxx)cos(π:Note (7 marks) (b) A uniform rod of length L is aligned along the x-axis between the points x=0 and x=L. The temperature v(x,t) at any point on a rod of length L at any instant is found by solving the partial differential equation 2 2 x vk t v ∂ ∂ = ∂ ∂ The ends x=0 and x=L are maintained at temperatures of 200C and 300C, respectively. The initial temperature distribution is given by v(x,0)=f(x). By using a substitution u(x,t)=v(x,t)-20- L 10x solve this partial differential equation. In particular find the solution when f(x)= L 10x . (13 marks) 10 Z-TRANSFORMS For a sequence f(n) the Z-Transform is defined by ∑ ∞ = −= 0n nf(n)zF(z) f(t) F(z) U(n)=1 1z z − Na az z − n 21)(z z − 2n 31)(z 1)z(z − + bne bez z − cosωn 1zcos2z )cosz(z 2 +− − ω ω sinωn 12zcos-z zsin 2 +ω ω f(n)a n       a zF nf(n) -zF(z) f(n+1) zF(z)-zf(0) f(n+2) zf(1)f(0)zF(z)z 22 −− 11 LAPLACE TRANSFORMS For a function f(t) the Laplace Transform of f(t) is a function in s defined by F(s) e f(t)dtst 0 = − ∞ ∫ where s>0. f(t) F(s) A=constant A s t N N! sN 1+ eat 1 s a− sinhkt k s k2 2− coshkt s s k2 2− sin tω ω ωs2 2+ cos tω s s2 2+ω e f(t)at F(s-a) ′f (t) sF(s)-f(0) ′′f (t) s F(s) sf(0) f (o)2 − − ′ 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 F(s)as− δ ( )t a− e-as Note: coshA e e 2 sinhA e e 2 A A A A = + = −− −