Math 2250 Lab 14
Name/Unid:
1. Consider the multiple mass-spring system shown in the figure below. We neglect the
curvature so that each mass is pulled in opposite directions by the springs that connect its
two neighbors. Recall that in an ordinary single mass-spring system that the restorative
force is proportional to the change in the length of the spring so that
mx00 = −kx
Imagine, now that there are two masses, m1 and m2 , connected by a single spring with
spring constant k. We then have the system
m1 x001 = −k(x1 − x2 )
m2 x002 = −k(x2 − x1 )
m1
x1
m5
x2
x5
x4
m4
x3
m3
m2
(a) Suppose that all 5 springs have the same spring constant, k. Use the above information to construct a system of differential equations for the system of the form
Mx00 = Kx
where
 
x1
 x2 
 

x=
 x3 
 x4 
x5
Specify the matrices M and K.
Page 2
(b) Let mi = 1 for i = 1, ..., 5. This reduces the above equation to x00 = Kx. Let
y = x0 . Convert the 5 × 5 system from part (a) to a 10 × 10 system for the vector
 
x1
 .. 
.

 
x
x 
z=
=  5
y
 y1 
.
 .. 
y5
This will have the form
z0 = Az
where A is a 10 × 10 matrix.
Page 3
(c) Suppose we have an 2n × 2n matrix
A=
B C
D E
where B, C, D, E are all n × n matrices. Then, if DE = ED, we have the result
that
det(A) = det(BE − CD).
Use this fact to show that if λ is an eigenvalue of the K in part (a), then ω =
is an eigenvalue of the matrix A from part (b).
Page 4
√
λ
(d) Find the eigenvalues of K. What are all the possible frequencies that the system can
oscillate at? Without actually computing the eigenvectors, write down the general
solution of the system in terms of linear combination of functions of t times the
eigenvectors. You may use any technology you want to find the eigenvalues.
Page 5
(e) Find a non-trivial equilibrium solution to the system in (a). How does this relate
to the eigenvector corresponding to λ = 0? Give a physical interpretation of this
result.
Page 6
2. Consider the network of tanks shown in the figure below.
r1,in = 3 [L/m]
c1,in = 20(1 + sin(2π ∗ t))
x1 (t)
V1 = 20
r3,1 = 1 [L/m]
r2,1 = 2 [L/m]
x3 (t)
V3 = 10
r2,3 = 1 [L/m]
x2 (t)
V2 = 10
r2,out = 3 [L/m]
(a) Let
 
x1

x = x2  ,
x3

c1,in
and cin = c2,in 
c3,in

denote the vector of the amount of dissolved salt in each tank and the vector of salt
concentrations going into each tank, respectively. Derive the system of differential
equations of the form
x0 (t) = Ax(t) + c(t)
(Use the next page if you need more space.)
Page 7
Page 8
(b) Find the eigenvalues and eigenvectors of A.
Page 9
(c) Write down the complementary solution xc that satisfies the homogeneous system.
Page 10
(d) Determine the particular solution xp .
Page 11
(e) Write the full general solution to the system in (a).
Page 12
3. An n × n matrix is called diagonalizable if there exists an invertible matrix S, such that
A = SΛS−1
where Λ is a diagonal matrix (that is Λij = 0 for all i 6= j).
(a) Show that for any integer k
Ak = SΛk S−1
Page 13
(b) Show that if


λ1
λ2


Λ=

..




.
λn
then



Λ =

k

λk1
λk2
Page 14
...
λkn




(c) Recall that the exponential of a matrix is defined as
∞
X
1 k
A .
e =
k!
k=0
A
Use this along with the results from parts (a) and (b) to show that

e λ1

eλ2
A
−1 
e =S 
..

.
Page 15

e λn


S

(d) The above shows that we can compute the matrix exponential if we know the
eigenvalues of the matrix, A, and the matrix S. It turns out that S is a matrix with
the eigenvectors of A as column vectors. If λi is an eigenvalue with corresponding
eigenvector vi then

S = v 1 · · · vn ,

λ1


Λ=

λ2


.

...
λn
That is, the eigenvectors in S are in the same order as the eigenvalues in Λ.
Use this to compute by hand eA when
A=
Page 16
1 2
2 1