SEHS4612 Numerical Methods for Engineers Assignment

Assignment Help Mechanical Engineering

Reference no: EM132501172

SEHS4612 Numerical Methods for Engineers Assignment - School of Professional Education and Executive Development - The Hong Kong Polytechnic University, Hong Kong

Question 1 - Consider the mass-spring system in the following figure.

The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying Mx¨ + kx = 0, which yields

Applying the guess x = x₀e^iωt as a solution, we get the following matrix:

Assume m₁ = m₂ = m₃ = 1 kg, and k = 2 N/m and let

(a) Use the Power method with initial guess v₀ = to solve the dominant eigenvalue and the corresponding eigenvector of matrix A.

(b) Use the dominant eigenvalue to solve for the frequency (ω).

Question 2 - The following data for the density of nitrogen gas versus temperature come from a table that was measured with high precision.

T, K	200	250	300	350	400	450
Density, kg/m³	1.708	1.367	1.139	0.967	0.854	0.759

(a) Use Newton's interpolating polynomials of order 1 through 3 to estimate the density at a temperature of 330 K. Choose the sequence of the points for your estimates to attain the best possible accuracy. That is, the points should be centered around and as close as possible to the unknown. What is your best estimate?

(b) Repeat part (a) using the Lagrange polynomial of order 1 through 3.

Question 3 - Given the following set of data

x	2	2.2	2.4	2.6	2.8	3
y	1.00	1.07	1.13	1.18	1.23	1.27

It is known that the data can be described closely by the model

1/y = (1/a)e^b/ax

(a) Use least-squares linear regression to find a and b by applying appropriate transformations.

(b) Compute the coefficient of determination of the model in part (a).

Question 4 - It is known that the function f (x) passes through the following three points:

(1.8, 10.89), (2.0, 14.78), (2.2, 19.86).

Use forward and backward difference approximations of O(h), and centred difference approximation of O(h²) to estimate the first derivative of f(x) at x = 2.0. If the true value of f (2.0) = 22.17, find the absolute error for each approximation. How will it affect the error for each case if the step size h changes from 0.2 to 0.1?

Question 5 - Consider the integral

I = ₀∫¹f(x)dx = ₀∫¹1/(1+x²)dx = π/4

(a) Apply the trapezoidal rule to approximate I on three data points f (0), f (0.5), f (1.0).

(b) Estimate the error if the trapezoidal rule are used on nine equally spaced data points.

(c) If trapezoidal rule is used to approximate I to the accuracy of at least 10^-7, how many divisions, n, should be used for the approximation?

Reference no: EM132501172

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