ME2016, Fall 2008, Dr. Ferri
Homework Assignment 12
Due Wednesday, December 3.
Problem 1. (5 points)
Consider the initial-value problem
2
2;(0)1ytyy
′
=
−= (1)
It can be shown that the exact solution to this nonlinear ode satisfying the initial condition is
2
() 1/( 1)yt t
=
+ (2)
(a) Perform Euler integration with h = 0.1 from 0 < t < 2 sec. Make a plot of the Euler solution and the
exact solution. Compute the maximum error over 0 < t < 2 sec.
(b) Repeat part (a) with h = 0.01. By what factor does the maximum error decrease? Does this match
your expectations?
(c) Integrate the solution to (1) using the 2nd-order Runge-Kutta method and h = 0.2. Plot the solution vs
the exact solution and vs the Euler answer from part (a). Based on the maximum error, which is more
accurate and by what factor?
(d) Integrate the solution to (1) from 0 < t < 2 using Matlab’s ode45 function (Type "help ode45" for
further info.) Plot the solution vs the exact solution. Compute the maximum error over 0 < t < 2 sec
and compare with the errors obtained using the methodology of parts (a) and ( c).
Problem 2. (5 points)
Find the step response of a spring-mass-damper system having
the following governing equation:
2
2()
dx dx
mckxFt
dt
dt ++=
; (0) (0) 0
dx
xdt
=
=
where F(t) is a unit step function given by
⎩
⎨
⎧
≤
>
=00
01
t;
t;
)t(F
Note that the problem is second-order, since the highest derivative in x is 2. Before proceeding, we need
to write an equivalent set of 2 first-order equations. Define y1(t) = x(t) and y2(t) = dx/dt. Then the system
of equations is:
)y,t(fy
m/)cyky)t(F(y
yy =⇒
−−=
=
212
21
(a) Find the step response from t = 0 to t = 50 seconds using the Matlab function ode45.m. Use the
parameter values m = 1kg, k = 1 N/m, c = 0.2 Ns/m. Your call will have the following form:
>> [T,X] = ode45(func,[0 50],[0 0]);
k
m
c
x(t)
F
