EAS439 Tutor-Marked Assignment Questions & Answers – SUSS, Singapore

Question 1

Figure Q1 shows the forces acting on an aircraft at level flight.

From Newton’s 2nd law, the governing equation is given by:

where and are the parameters for the aircraft aerodynamic drag polar, is the wing area, and is the air density.

(a) Formulate numerical approximations to the equation using forward difference to compute the aircraft speed V, for the level flight to accelerate speed from V1 to V2.

(8 marks)

(b) An aircraft has the following characteristics:

Based on the numerical approximation formulated in part (a), compute the aircraft speed V at different times, using Δt = 5. Copy and fill in Table 1b to answer the question.

Table Q1b

(9 marks)

(c) Analyse the accuracy and efficiency of the numerical formulation used, by commenting the effect of Δt on the result accuracy and the algorithm efficiency.

(3 marks)

(d) Explain the difference between “Total Error” and “Approximation Error”, and their respective usage.

(5 marks)

Question 2

(a) Given the following system of equations:

.

Calculate the solution using Gauss-Jordan Elimination method. Show all steps of the computation.

(10 marks)

(b) The deflection of a cantilever beam under the uniformly distributed load ( ) is given as follows:

,

where y is the vertical deflection, E is the modulus of elasticity, I is the area moment of inertia, and L is the length of the beam. The cantilever beam is fixed at root x = 0 and the free end is at x = L. Use Simpson’s 3/8 rule to design and implement a computational step to calculate the deflection of the beam at x = L.

(10 marks)

Question 3

The measured data for the drag coefficient Cd of a symmetric NACA-0015 airfoil at different angles of attack α is given in Table Q3 below.

Table Q3

(a) By least-squares, perform a polynomial regression of the function below to the given data and compute parameters a, b c and d. Plot the measured data and Cd(α) to demonstrate the fit. Show all steps of your working clearly.

(10 marks)

(b) From a nonlinear regression of the data, a new function of the drag coefficient is obtained as

.

Using the bisection method, compute the angle of attack at the drag coefficient of 0.75 using the new function. Use an initial interval of [30, 40] and a stopping criterion of εs = 0.1%. Copy and fill in Table Q3b to answer the question (use 4 d.p. for all entries).

Table Q3b

(10 marks)

Question 4

An aluminum rod that is insulated at all points except at its ends experiences heat conduction along its length as illustrated in Figure Q4 below.

The temperature T(x, t) along the rod is governed by the 1D heat equation below, where α is the thermal diffusivity of the rod’s material.

(a) Formulate the numerical solution of the heat equation using the explicit method.

(3 marks)

(b) By discretizing the rod into 6 equal segments and using a time step of 60 seconds, calculate the approximate solutions of T(x, t) in °C (round off to 1 decimal place). Copy and fill in Table Q4b to answer the question. The initial temperature of the rod is 45°C and the left and right ends are maintained at 25°C and 230°C respectively. The properties of the aluminum rod are given below.

Length = 3 m. Density = 2810 kg/m3.

Thermal conductivity = 235 W/m.K. Specific heat = 900 J/kg.K.

Table Q4b

(10 marks)

(c) By direct integration, solve for the exact steady-state temperature T(x) of the aluminum rod. Hence, appraise your numerical solution in part (b) at t = 30 hrs and analyse its accuracy.

(7 marks)

(d) The left end (x = 0) of the aluminum rod is now thermally insulated and no longer subjected to a fixed temperature. In addition to the numerical solution in part (a), formulate the extra condition/s needed to simulate the temperature of the rod subjected to an insulated left end.

(5 marks)

(e) For the aluminum rod with its left end thermally insulated, calculate the approximate solutions of T(x, t) in °C (round off to 1 decimal place). Copy and fill in Table Q4e to answer the question. Use all other conditions given in part (b) except for the rod’s left end temperature.

Table Q4e

(10 marks)