MATH1145: You are asked to calculate the distance covered by the vehicle in the time between 0 and 10 seconds both by analytical integration (exact solution) and by 5 different numerical integration approximations: Engineering Mathematics Coursework, UOG, UK

MATH1145 – Engineering Mathematics II

Mathematical analysis and MATLAB modelling coursework Mechanical Engineering, Academic year: 2020-2021

Use the template provided on Moodle to prepare your submission. Submission Deadline: Upload your individual reports by 2nd April 2021, 11:30 pm

For more information about the Rubric, refer to the Assessment details document on Moodle.

TASK 1: numerical integration and differentiation tasks (40%)

a) The velocity of a vehicle, for 0 ≤ 𝑡 ≤ 10𝑠𝑒𝑐 is described by the function:
𝑣(𝑡) = 50 + (𝑐 + 1) × 𝑡 × cos (𝑡) (km/h), where c is the last digit of your student ID. You are asked to calculate the distance covered by the vehicle in the time between 0 and 10 seconds, both by analytical integration (exact solution) and by 5 different numerical integration approximations:

i) The 3 Riemann sums (left, right & midpoint)

ii) The trapezoidal rule, and

iii) The Simson’s Rule, and demonstrate how each of these methods converges to the exact solution, as you increase the number of the sub-intervals you will use to discretise the time interval t[0,10]. Reflect on the results.

b) Insert the discrete acceleration time history ‘THS_0c’ (where c is the last digit of your student ID) in MATLAB and apply any valid numerical integration method to calculate the velocity and displacement time histories. Then apply any numerical differentiation technique to the displacement time history to go back to velocity and acceleration. Comment on the results or any discrepancies you observe.

Extra task: Try to use less discrete points for the displacement time history (re-sample, e.g. one every 2, every 4 or every 5 points), and differentiate again. Compare and reflect on the results.

TASK 2: Application of ODEs into the analysis of a beam (60%)

Consider a beam with an inverted tee-shaped cross-section which is made of stainless steel (𝐸 = 210𝐺𝑃𝑎) and it is fixed in one end and supported on a pin on the other end, as shown in the figure. The beam is subject to a uniformly distributed load 𝑞0 along its whole length.
a) Develop the ODE of the problem. Solve the ODE to get the analytical solution.
b) Solve the ODE applying a Finite differences stencil, until you achieve satisfactory agreement with the analytical solution. Show the formulation of the equations and program them in MATLAB. Select any method you prefer to solve the linear system of equations.
c) Write a MATLAB code that locates the centroid axes of the cross-section
d) Write a MATLAB code that determines the moment of inertia of the cross-section
e) Apply a suitable numerical analysis to calculate the bending moment and shear force diagrams along the axis of the beam
f) Write a MATLAB code to determine the maximum bending moment and for this moment calculates the stress at points K and H in the cross-section

Cross-section of the beam

Table 1 Parameters for your analysis

Answer

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