Initial Problem Solving Lab Example

The following provides an example for the initial version of the Problem Solving Lab Assignments in EGR 312. The problem explained and described below refers to the "falling parachutist" problem that is described in the course text book (Numerical Methods for Engineers 8th Edition - Chapra and Canale) and in class. Please use this as a guide when completing your initial problem solving lab as it includes all key elements that you are expected to complete as part of your submission. Note that your initial assignment does not need to be in the form of a webpage. Remember that you are encouraged to present your results in a way that seems most appropriate to your problem and you you should not exactly replicate the format and content you see here. The content below was developed by Dr. Di Vittorio and Nick Corak (EGR 312 TA). An example of the Final Problem Solving Lab is also provided.

Introduction

This problem explores the concept of terminal velocity through the modeling of the velocity of a falling parachutist before they release their parachute. The parachutist has a mass of 68.1 kg and their initial velocity is assumed to be equal to zero. The drag coefficient is assumed constant and equal to 12.5kg/s.

In this example, a free body diagram is used to set up the model in terms of acceleration, or the derivative of velocity. The model takes the form of a differential equation with an analytical solution, so the “true” solution can be calculated. For the numerical solution, Euler’s method will be applied, which is a first order method for solving differential equations.

Conceptual Questions & Response

This section will be specific to your Problem Solving Lab Assignment, where you are expected to respond to big-picture questions regarding the numerical method you are learning in class and the problem-solving approach

Mathematical Model

First, a mathematical model needs to be formulated to describe the net forces on the parachutist. The result is a differential equation describing velocity change over time, as shown in Figure 2. In these equations, the following parameters are used:

F = external forces (N) (Fg = gravitational force; Fd = drag force)

m = mass (kg)

a = acceleration (m/s^2)

v = velocity (m/s)

c_d = drag coefficient

g = gravity (m/s^2)

Note: Sometimes you will be provided with the mathematical model and will not need to show the full derivation, but you should at least explain the model and how it was derived. You can lump this in with "Numerical" or "Alternative" or you can place it in its own section.

Figure 1: Mathematical Model

Numerical Approach

Euler’s method can be applied to solve the differential equation numerically. Euler’s is a first order approximation to the derivative and can be applied to our problem according to Figure 2, where "i" is the iterator for time (t).

Figure 2: Explanation of Numerical Approach

Sample Calculations

Sample calculations for Euler's method applied to the falling parachutist problem are provided in Figure 3, starting with a velocity = 0 at time = 0. A time step of 1 second was used for three iterations. The calculations show that velocity is still increasing after 4 seconds, although the rate of increase appears to be slowing down. The simulation will have to be extended to reach terminal velocity, but this can be done most efficiently with a computer.

Pseudocode

Define constant parameters

g = 9.81

c = 12.5

m = 68.1

Calculate true velocity using analytical solution and plot over time

Set initial conditions

t0 = 0 (time)

v0 = 0 (initial velocity)

h = 1 (step size)

Begin FOR loop - go out to 150 seconds

approximate v(t+1) using Euler's (refer to sample calcs)

store time and velocity in variable for each step size

END loop

Calculate true error and approximate error

comparison plot of true and approximate velocity for different step sizes

Plot error for different step sizes together

Figure 3: Sample Calculations for Numerical Approach

Analytical Approach

Instead of simulating the velocity using Euler's method, the exact equation for velocity can be derived by taking the integral of the ODE, using concepts and strategies from calculus. The analytical derivation is shown in Figure 4. Sample hand calculations using the resulting analytical equation are shown in Figure 5 below. These sample calculations can be directly compared to the numerical since the initial conditions and time step are the same.

Figure 5: Sample Calculations for Analytical Approach

Figure 4: Analytical Derivation of Velocity

Expectations For Results

I expect that the velocity will increase until it reaches the terminal velocity. See the sketch below. Based on the calculations of the numerical and analytical solutions, it looks like the numerical solution is providing results that are slightly larger than the analytical solution. The analytical solution can be found to be the velocity as time goes to infinity which I calculated as 53.4 m/s.

Figure 6: Expectations for Results

Error - Sample Calculation and Explanation

I could calculate the absolute percent error at each timestep by comparing the approximation of velocity (using Euler’s method) to the true value calculated with the analytical solution of the differential equation. Sample hand calculations are shown below. Observe that the true percent relative error is decreasing as time increases. If I changed the step size, I expect that error would also change and should be smaller for a smaller step size since the equation would be more linear in a smaller segment.

Figure 7: Sample Error Calculations

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