[Solved] MAE 3210 Homework 6

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:
   • If n is odd, it applies the multiple-application trapezoidal rule in

each dimension to approximate.

• If n is even, it applies the multiple-application Simpsons 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:

• Analytically, to obtain a true value.
• 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) Eulers method and(b) the standard 4th order Runge-Kutta method, for solving a given first-order one-dimensional 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 F_d (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 c_d (kg/m).
   • Assume that a falling object has mass m = 100 (kg) with a drag coefficient of c_d = 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 Newtons second law, explain the derivation of the following ODE for the downward velocity v = v(t) of the falling object:

. (1)

• 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
  • Eulers method.
  • 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 t_f when the height y of the falling object reaches zero, i.e. when y(t_f) = 0. There are two ways to solve this problem.

1. 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(t_f) = 0.
2. 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(t_f) = 0.

4. Write code for two separate algorithms to implement (a) Eulers 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 (Ns/m).

• 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.
• Assume that the mass is m = 10 kg, the stiffness k = 12 N/m, the damping coefficient is c = 3 Ns/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),
  • using Eulers method with step size h = 0.5, and then with step size h = 0.
  • using the standard 4th order Runge-Kutta method with step sizeh = 0.5, and then with step size h = 0.
• Assume that the mass is m = 10 kg, the stiffness k = 12 N/m, the damping coefficient is c = 50 Ns/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),
  • using Eulers method with step size h = 0.5, and then with step size h = 0.
  • using the standard 4th order Runge-Kutta method with step sizeh = 0.5, and then with step size h = 0.