Assignment 6: Matlab Functions Problem 1

CEE 3804: Computer Applications in Civil Engineering Spring 2011 

Date Due: Solutions 

Problem 1 

Assignment 6: Matlab Functions 

Instructor: Trani 

A civil engineer is designing a rainstorm water management system for a shopping mall. During a severe thunderstorm, the 

water runoff generated by the large parking lot at the shopping mall is given by the function: 

runoff = k 2 

+ k 1 

sin(t / k 3 

)e (−t /k 4 ) 

Where runoff is the runoff volume (cubic meters per second) generated by the parking lot, t is the time (in seconds) after the 

thunderstorm starts and k 1 

through k 4 

are parameters of the runoff function. 

Task 1 

Create a Matlab function to calculate the runoff for a given value of time t. As part of the input variables to the function runoff, 

include the four input parameters k1 through k4 for a 100 year storm with numerical values as follows: 

k1 = 50; 

k2 = 2; 

k3 = 1500; 

k4 = 800; 

________________________ 

% Function to estimate runoff volume at a shopping mall 

function [runoff] = function_Runoff(k1,k2,k3,k4,t) 

% runoff is the runoff volume (cubic meters per second) 

% generated by the shopping mall into the ponding area 

% t is the time (in seconds) after the thunderstorm starts 

% k1, k2, k3 and k4 are model coefficients. 

runoff = k1 * sin(t/k3) .* exp(-t/k4) + k2; 

________________________ 

Task 2 

Create a Matlab script to call the function created in Task 1. Test the function for values of time (t) starting at t=0 and ending at 

t=4500 seconds. Plot the runoff as a function of time. Label appropriately. 

In your plot use the following plot attributes: 

a) Marker of plotted values = square 

b) Line type = dashed black line (10 point) 

________________________ 

% Function to call the function runoff volume at a shopping mall 

CEE 3804 Trani Page 1 of 13

% Ponding volume at a shopping mall 

% Calls function Runoff 

clc 

clear 

% qRunoff is the runoff volume (cubic meters per second) 



k1 = 50; 

k2 = 2; 

k3 = 1500; 

k4 = 800; 

tLast = 4500; 

% multiplicative parameter of the function 

% additive parameter of the function 

% parameter of sinusoidal term 

% parameter of exponential term 

% final time to do the calculation 

t=1:1:tLast; % create an array with values of time 

qRunoff(t) = function_Runoff(k1,k2,k3,k4,t); 

% Make a nice plot 

figure 

plot(t,qRunoff,'o') 

xlabel('Time (seconds)') 

ylabel('Volume (cu. meters/second)') 

grid 

% Make a plot of the derivative of runoff 

derQin = gradient(qRunoff); 

figure 

plot(t,derQin,'^') 


ylabel('Gradient of qRunoff (cu. meters/second^2)') 

grid 


Figure 1. Runoff Function. 

_______________________ 

Task 3 

Modify the Matlab script created in Task 2 to find the area under the curve of the runoff as a function of time. Use the Matlab 

“quad” function to estimate the integral. Plot the results and output the result of the area under the curve in the command window 

using the “disp” function in Matlab. How much runoff is produced in the 4500 second storm event? State the units of the answer. 

__________________________ 

% Script to calculate the area under the runoff function 

% Programmer: T. Trani 

global k1 k2 k3 k4 

% Calls function function_Runoff_rev 

% areaUnderRunoff is the runoff volume (cubic meters per second) 



k1 = 50; 

k2 = 2; 

k3 = 1500; 

k4 = 800; 

tLast = 4500; 

% multiplicative parameter of the function 

% additive parameter of the function 

% parameter of sinusoidal term 

% parameter of exponential term 

% final time to do the calculation (seconds) 


% Estimate the area under runoff function 

areaUnderRunoff = quad('function_Runoff_rev',0,tLast); 

disp(['Area under the Curve is ',num2str(areaUnderRunoff)]) 

______________________ function_Runoff_rev.m function ________ 

% Function to estimate runoff volume at a shopping mall 

function [runoff] = function_Runoff_rev(t) 


% runoff is the runoff volume (cubic meters per second) 



runoff = k1 * sin(t/k3) .* exp(-t/k4) + k2; 

________________________________ 

Area under the Curve is 25652.45 cubic meters 

Task 4 

Estimate the dimensions of a ponding volume needed to store all the runoff volume generated by the 100 storm event. State 

dimensions of the ponding volume (base x width x height). 

One estimate of the ponding volume would be 100 x 100 x 2.57 meters. This is equivalent to two Football stadiums with 

a depth of 2.27 meters (8.5 feet). 



A train engineer collects data during the certification of the new high-speed train to be introduced in the Northeast Corridor in the 

United States. The data collected records train acceleration (a) vs. velocity (V). The data is presented in the table below. 

Table 1. Train Acceleration and Speed Data. 

Train Velocity (m/s) 

Maximum Train Acceleration 

(m/s 2 ) 

0.00 3.51 

21.31 2.32 

30.40 1.73 

50.16 0.94 

69.65 0.39 

81.45 0.12 

94.20 0.01 

Task 1 

Create a Matlab script to plot the data and (in the same script) to find the best second and third order polynomials that fit the data 

(i.e., use the “polyfit” command). Save the coefficients of the polynomials and evaluate 100 data points of these polynomials 

across a range of speeds from 0-95 m/s (the usable operating speeds for this train). Compare with the original data. 

____________________________ Matlab Script for Task 1, Problem 2 ________________ 

% Problem 2 

% Task 1 

% Data 

A=[0.00 3.51;21.31 2.32; 30.40 1.73; 50.16 0.94; 69.65 0.39; 81.45 0.12; 94.20 0.01]; 

velocity=A(:,1); 

maxAcceleration=A(:,2); 

%% fitting to second order polynomial 

[p2,s2]=polyfit(velocity, maxAcceleration, 2); 

%% fitting to third order polynomial 

[p3, s3]=polyfit(velocity, maxAcceleration, 3); 

%% 100 data points for evaluating the two polynomials 

X=linspace(0,95,100); 

%% evaluating of second-order polynomial 

Y2=polyval(p2,X); 


%% evaluating of third-order polynomial 

Y3=polyval(p3,X); 

% Make a good plot. We use the plot command with a variation (h = plot) to embellish the plot using 

% Matlab handle graphics 

h=plot(velocity,maxAcceleration,'ok',X,Y2,'b-',X,Y3,'r-'); 

set(h, 'linewidth',2); 

set(gca, 'fontsize',20); 

title('Velocity-Maximum Acceleration Graph','fontsize',20); 

xlabel('Velocity (m/s)', 'fontsize',20); 

ylabel('Maximum Acceleration (m/s^2)', 'fontsize',20); 

legend('Observed data','Second-Order Polynomial','Third-Order Polynomial'); 

grid on 

_____________________________________ 

The Matlab script produces the graph shown in Figure 2. 

Figure 2. Plot of Train Acceleration Data vs. Speed. 

Task 2 

Using the best fit polynomials estimated in Task 1, evaluate the goodness of fit of both answers looking at the residuals or the 

local differences between the data points and the fitted polynomials. Select the best model fit. Comment on your solution. 

The script to find the residuals is: 


%% task 2 

%% finding residuals 

normOfResiduals2=s2.normr; 

normOfResiduals3=s3.normr; 

Comparing normOfResiduals2 (2nd order polynomial) with norm of residuals for the third-order 

polynomial we conclude that the 3rd-order polynomial is closer to the data and hence a better 

model. 

The best fit equation of the 3rd-order polynomial is: 

a(t) = dV = k 1 

+ k 2 

V + k 3 

V 2 + k 4 

V 3 

dt 

where: 

k 4 

= 0.000000576703820 

k 3 

= 0.000235555589820 

k 2 

= -0.064571096680574 

k 1 

= 3.520152968080940 

Task 3 and Problem 3 (Task 1) 

Create another set of scripts (as needed) and using the best polynomial fit found in Task 2, solve the differential equation of 

motion for the train using the Matlab ODE45 solver. The equation of motion to estimate acceleration of the train is of the form: 

a(t) = dV 

dt 

= k 1 

+ k 2 

V + k 3 

V 2 + k 4 

V 3 

where: a(t) = dV is the instantaneous train acceleration (m/s 2 ), V is the train speed (m/s) and parameters k 1 

,k 2 

,k 3 

,k 4 

dt 

are the coefficients of the polynomial found in Task 2. 

Solve the differential equation with initial conditions V 0 

and a final time of 120 seconds. Plot the velocity and acceleration 

profiles in a 2 x1 window using the Matlab subplot command. 

_______________ Main Matlab Script to Estimate Train Speed and Distance Profiles ___________ 

% Main file to find speed and distance traveled profile for a high-speed 

% rail system 

% This functions calls: trainFunction.m which contains rates of change of the velocity and distance state 

% variables of the train as a function of time 

% Define the following: 

% yprime(1) = acceleration 

% yprime(2) = speed 

% yprime(1) = k1 + k2 * y(1) + k3 * y(1)^2 + k4 * y(1) ^3 

% yprime(2) = y(1) 


% y(1) = velocity (m/s) 

% y(2) = distance traveled (m) 


% Define Initial Conditions of the Problem 

yo = [0 0]; 

to = 0.0; 

tf = 120; 

% yo are the initial speed and distance traveled 

% to is the initial time to solve this equation 

% tf is the final time 

% define constants k1 - k4 (obtained from regression analysis) 

k1 = 3.520152968080940; 

k2 = -0.064571096680574; 

k3 = 0.000235555589820; 

k4 = 0.000000576703820; 

tspan =[to tf]; 

[t,y] = ode45('trainFunction',tspan,yo); 

% call ODE solver 

% Plot the results of the numerical integration procedure 

subplot(2,1,1) % plots time-distance traveled profile 

h1 = plot(t,y(:,2),'^--b'); % gets handle graphics 

set(h1, 'linewidth',1); % sets the line width to 1 

set(gca, 'fontsize',20); % sets the font size to 20 


ylabel('Distance Traveled (m)') 

grid 

subplot(2,1,2) % plots time-speed profile 

h2 = plot(t,y(:,1),'o--r') 




ylabel('Speed (m/s)') 

grid 


______________ Function trainFunction.m _____________________ 

function yprime=trainFunction(t,y) 

% Define the following: 

% yprime(1) = acceleration 

% yprime(2) = speed 

% yprime(1) = k1 + k2 * y(1) + k3 * y(1)^2 + k4 * y(1) ^3 

% yprime(2) = y(1) 

% y(1) = velocity (m/s) 

% y(2) = distance traveled (m) 


yprime(1) = k1 + k2 * y(1) + k3 * y(1).^2 + k4 * y(1).^3; % finds acceleration 

yprime(2)= y(1); 

yprime = yprime'; 

_______________________________ 

Execution of the Main file in Matlab produces: 


Figure 3. High-Speed Train Distance and Speed Profiles. Note: the train reaches 80 m/s (180 mph) in around 100 seconds after 

start. 



Task 2 

Solve the train problem using Simulink. Create a Simulink model to estimate the acceleration, velocity and distance profiles of 

the train as a function of time using the best fit polynomial model found in Task 2 of Problem 2. 

Equation to be modeled in Simulink. 

a(t) = dV 

dt 

= k 1 

+ k 2 

V + k 3 

V 2 + k 4 

V 3 

Figure 4. Simulink Model of the High-Speed Train. Third-Order Polynomial. 


Task 3 

Export the results obtained by the Simulink model created in Task 2 and plot in Matlab. Label all the axes as needed. Simulink 

plots are not acceptable. 

__________________ Matlab Script to Plot Simulink Results obtained from the Workspace _______ 

% Script to plot the results of a simulink model 

% Simulink model outputs: 

% tout = time signal (seconds) for state variables (speed and distance) 

% xout = state variables (speed and distance) 

% xout(:,1) = speed (m/s) 

% xout(:,2) = distance (m) 

% Acceleration = output of acceleration time trace (m/s-s) 

! % NOTE: Acceleration by default is a structured array 

! % Plot Acceleration.signals.values 

subplot(3,1,1) % plots time-acceleration profile 

h3 = plot(tout,Acceleration.signals.values,'*--k') 




ylabel('Acceleration (m/s-s)') 

grid 

subplot(3,1,2) % plots time-speed profile 

h2 = plot(tout,xout(:,1),'o--r') 




ylabel('Speed (m/s)') 

grid 

subplot(3,1,3) % plots time-distance traveled profile 

h1 = plot(tout,xout(:,2),'^--b'); % gets handle graphics 




ylabel('Distance Traveled (m)') 

grid 


Figure 5. Plot of Acceleration, Speed and Distance Traveled by High-Speed Train.

Assignment 6: Matlab Functions Problem 1

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