MAE3210-Homework 6 Solved

1.      (a) Develop an algorithm which, for a given function of two variables f(x,y), interval bounds a and b with a < b, and c and d with c < d, and input integer n ≥ 1, does the following:

(i)     If n is odd, it applies the multiple-application trapezoidal rule in

each dimension to approximate .

(ii)   If n is even, it applies the multiple-application Simpson’s 1/3 rule

in each dimension to approximate .

(b) Suppose the temperature T (oC) at a point (x,y) on a 16 m2 rectangular heated plate is given by

T(x,y) = x2 − 3y2 + xy + 72,

where −2 ≤ x ≤ 2 and 0 ≤ y ≤ 4 (here x and y are measured in meters about a reference point at (0,0)). Determine the average temperature of the plate:

(i)     Analytically, to obtain a true value.

(ii)   Numerically, using the algorithm you developed in question 1(a) above, and plot the true percent relative error t as a function of n for 1 ≤ n ≤ 5. Provide some interpretation of the results.

2.      Write code for two separate algorithms to implement (a) Euler’s method and(b) the standard 4th order Runge-Kutta method, for solving a given first-order one-dimensional ODE. Design the code to solve the ODE over a prescribed interval with a prescribed step size, taking the initial condition at the left end point of the interval as an input variable.

3.      The drag force Fd (N) exerted on a falling object can be modeled as proportional to the square of the objects downward velocity v (m/s), with a constant of proportionality cd (kg/m).

(a)    Assume that a falling object has mass m = 100 (kg) with a drag coefficient of cd = 0.25 kg/m, and let g = 9.81 (m/s2) denote the constant downward acceleration due to gravity near the surface of the earth. Starting from Newton’s second law, explain the derivation of the following ODE for the downward velocity v = v(t) of the falling object:

                                                                              .                                            (1)

(b)    Suppose that this same object is dropped from an initial height of y0 = 2 km. Determine when the object hits the ground by solving the ODE you derived in question 3(a) using

(i)     Euler’s method.

(ii)   the standard 4th order Runge-Kutta method.

HINT: Note that, with the velocity v oriented downward, the height y = y(t) satisfies  . You are asked to find the final time tf when the height y of the falling object reaches zero, i.e. when y(tf) = 0. There are two ways to solve this problem.

A.     You can use your algorithm for solving one-dimensional ODEs (Eulerand Runge-Kutta 4) from question 2 to solve the ODE (1) to find v = v(t) (at discrete time points) with initial condition v(0) = 0. Then, you can use your one-dimensional ODE algorithms, again, to solve   with initial condition y(0) = 2000 m, and try to identify when y(tf) = 0.

B.     Alternatively, you can use your algorithm for solving two-dimensionalODEs (Euler and Runge-Kutta 4) from question 4 to solve the coupled

ODE system

 ,

with initial condition y(0) = 2000, v(0) = 0. Then, try to identify when y(tf) = 0.

4.      Write code for two separate algorithms to implement (a) Euler’s method and(b) the standard 4th order Runge-Kutta method, for solving a given first-order two-dimensional system of ODEs. Design the code to solve the system of ODEs over a prescribed interval with a prescribed step size.

5.      The motion of a damped mass spring is described by the following ODE

                                                                       ,                                                 (2)

where x = displacement from equilibrium position (m), t = time (s), m = mass (kg), k = stiffness constant (N/m) and c = damping coefficient (N·s/m).

(a)    Rewrite the 2nd order ODE (2) as a two-dimensional system of first orderODEs for the displacement x = x(t) and velocity v = v(t) of the mass attached to the spring.

(b)    Assume that the mass is m = 10 kg, the stiffness k = 12 N/m, the damping coefficient is c = 3 N·s/m, the initial velocity of the mass is zero (v(0) = 0), and the initial displacement is x = 1 m (x(0) = 1). Solve for the displacement and velocity of the mass over the time period 0 ≤ t ≤ 15, and plot your results for the displacement x = x(t),

(i)     using Euler’s method with step size h = 0.5, and then with step size h = 0.01.

(ii)   using the standard 4th order Runge-Kutta method with step sizeh = 0.5, and then with step size h = 0.01.

(c)    Assume that the mass is m = 10 kg, the stiffness k = 12 N/m, the damping coefficient is c = 50 N·s/m, the initial velocity of the mass is zero (v(0) = 0), and the initial displacement is x = 1 m (x(0) = 1). Solve for the displacement and velocity of the mass over the time period 0 ≤ t ≤ 15, and plot your results for the displacement x = x(t),

(i)     using Euler’s method with step size h = 0.5, and then with step size h = 0.01.

(ii)   using the standard 4th order Runge-Kutta method with step sizeh = 0.5, and then with step size h = 0.01.