Math 2280-1 Maple Project 3
Summer 2010
Directions: Hand in a single Maple document, printed and stapled neatly, which contains the
answers to the exercises below. At the top of this document, you should create a text field with an
appropriate title, the date, your name, and UID. Below this header, please answer the exercises in
order. If an exercise calls for computations by hand, you can type up your solutions in a text field,
or leave enough space so that you can hand-write your computation/explanation after printing the
document.
Maple Help: Maple has an extensive Help menu! You can search for commands to find syntax,
related commands, and extensive examples. Additionally, there are some introduction to Maple
guides posted on our class webpage (www.math.utah.edu/ kitchen/2280Sum10). You can also get
help from tutors in the tutoring center, and follow the guidelines in the book as necessary.
Exercises, Part 1:
You may be aware of the construction at Marriott the last few years. One objective of the construction is listed as “Improve seismic stability and bring the building up to code.” In short, the
building has recently been “earthquake-proofed”, but what does this mean? What is the mathematics behind “the code” that Marriott failed to satisfy before? Why should we be more confident
now that the library won’t collapse in an earthquake?
In this part of the project, you will study a non-homogeneous second order system which describes
the oscillations of a 4-story building (all stories above ground) experiencing an earthquake. This
project roughly follows section 5.3 Application of the text on pages 330-332. Suppose each of
the 4 stories weighs 32 tons, so the mass of each is m = 2000 slugs. Also, assume a horizontal
restoring force of k = 5 tons per foot (=10,000 lbs/ft) between the adjacent floors. Therefore, if
xi (t) describes the horizontal displacement of the ith floor, the forces exerted by the adjacent floors
on the ith floor satisfy the differential equation (depending on i)
mx001
mx002
mx003
mx004
=
−kx1 + k(x2 − x1 )
= −k(x2 − x1 ) + k(x3 − x2 )
= −k(x3 − x2 ) + k(x4 − x3 )
=
−k(x4 − x3 )
= −2kx1 + kx2
=
kx1 − 2kx2 + kx3
=
kx2 − 2kx3 + kx4
=
kx3 − kx4
(1)
Here, x1 is the ground floor, with the −kx1 term corresponding to the restorative force exerted by
the foundation of the building, and x4 is the top floor. Let x = (x1 , x2 , x3 , x4 ) be the (horizontal)
position vector of the floors. We can write the above system of second order equations as a single
k
Ax, where kA is the coefficient matrix for the right hand side
matrix equation of the form x00 = m
of the equation. With the values given above for k and m, we have


 
−2 1
−10
5

  5
k
10, 000 
−10
5
=
 1 −2 1

(2)
A=
1 −2 1  
5
−10 5 
m
2000 
5
−5
1 −1
1
Problems:
k
1. Use Maple to find the eigenvalues of the matrix m
A. Make a table (by hand or in Maple) like
the one on page 331
√ of our text to display your four eigenvalues (λ) and the corresponding
frequencies (ω = −λ) and periods (P = 2π
ω ). Arrange the table so the periods are listed in
increasing order. These are the frequencies and periods of natural oscillations of the building.
2. Suppose an earthquake produces a horizontal oscillation of the ground, described by the
function 14 cos ωt. Therefore, the inertial force exerted on each floor by the earthquake is
m 2
4 ω cos ωt, and consequently, the motion of our building undergoing this earthquake is modelled by the nonhomogeneous system
x00 =
1
k
Ax + ( ω 2 cos ωt)b
m
4
(3)
where b = (1, 1, 1, 1). Assume a particular solution vector takes the form xp (t) = c cos ωt for
some constant vector c. Use the method of undetermined coefficients to determine c, if the
period ( 2π
ω ) of the earthquake oscillations is 2 seconds. You will want to use Maple to solve
for c by setting up an equation similar to (32) on page 327 of the text, then solving that
linear system using Maple. What happens if the period of the earthquake oscillations is 2.8
seconds?
3. For ω arbitrary,
again solve for c, in terms of ω. If we write the entries of c as c = (c1 , c2 , c3 , c4 ),
p
let c := c21 + c22 + c23 + c24 . This will be a function of ω. Plot c(ω). Observe the spikes in
your plot correspond to the periods you found in problem 1. They indicate the vibrations
of the earthquake are producing a type of resonance with respect to the natural modes of
oscillation for your building. In these instances, the building is most susceptible to damage.
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Exercises, Part 2:
These exercises follow section 6.3 Application on page 412 of the text. You own a large wildlife
conservation preserve that you originally stocked with F0 foxes and R0 rabbits on January 1, 2007.
The following differential equations model the number of foxes F (t) and rabbits R(t) after t months:
dR
dt
dF
dt
= (0.01)pR − (0.0001)aRF,
= −(0.01)qF + (0.0001)bRF,
where p and q are the two largest digits (with p < q) and a and b are the two smallest nonzero
digits (with a < b) in your student ID number.
Problems:
1. Graph the phase plane portrait for your predator-prey system. Be sure to label the graph
and your axes.
2. Using your phase plane portrait, choose R0 and F0 – perhaps several hundred of each – so
that the resulting solution curve in the RF -plane is a fairly eccentric closed curve. Note
eccentricity increases when the R0 is large relative to the F0 .
3. Plot the solution curves for R(t) and F (t) on the same graph and use this graph to determine
the period of oscillation of each population, the maximum and minimum number of rabbits
and the calendar dates on which they first occur, and the max and min number of foxes and
the calendar dates on which they first occur. Display 5 graphs: one showing the at least the
first max/min of rabbits/foxes and one each zoomed in on the max/min of rabbits/foxes to
justify your answers.
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