Page 92 - Matrix Analysis & Applied Linear Algebra

86 Chapter 3 Matrix Algebra
Example 3.2.1

Consider two springs that are connected as shown in Figure 3.2.1.

Node 1 k 1 Node 2 k 2 Node 3

x 1 x 2 x 3

F 1 -F 1 -F 3 F 3
Figure 3.2.1
The springs at the top represent the “no tension” position in which no force is
being exerted on any of the nodes. Suppose that the springs are stretched or
compressed so that the nodes are displaced as indicated in the lower portion
of Figure 3.2.1. Stretching or compressing the springs creates a force on each
18
node according to Hooke’s law that says that the force exerted by a spring
is F = kx, where x is the distance the spring is stretched or compressed and
where k is a stiffness constant inherent to the spring. Suppose our springs have
stiffness constants k 1 and k 2 , and let F i be the force on node i when the
springs are stretched or compressed. Let’s agree that a displacement to the left
is positive, while a displacement to the right is negative, and consider a force
directed to the right to be positive while one directed to the left is negative.
If node 1 is displaced x 1 units, and if node 2 is displaced x 2 units, then the
left-hand spring is stretched (or compressed) by a total amount of x 1 −x 2 units,
so the force on node 1 is
F 1 = k 1 (x 1 − x 2 ).
Similarly, if node 2 is displaced x 2 units, and if node 3 is displaced x 3 units,
then the right-hand spring is stretched by a total amount of x 2 − x 3 units, so
the force on node 3 is
F 3 = −k 2 (x 2 − x 3 ).
The minus sign indicates the force is directed to the left. The force on the left-
hand side of node 2 is the opposite of the force on node 1, while the force on the
right-hand side of node 2 must be the opposite of the force on node 3. That is,

F 2 = −F 1 − F 3 .
18
Hooke’s law is named for Robert Hooke (1635–1703),an English physicist,but it was generally
known to several people (including Newton) before Hooke’s 1678 claim to it was made. Hooke
was a creative person who is credited with several inventions,including the wheel barometer,
but he was reputed to be a man of “terrible character.” This characteristic virtually destroyed
his scientific career as well as his personal life. It is said that he lacked mathematical sophis-
tication and that he left much of his work in incomplete form,but he bitterly resented people
who built on his ideas by expressing them in terms of elegant mathematical formulations.