Understanding Engineering Mathematics

2.Systems of forces acting on a body in equilibrium can also lead to systems of linear
equations. For example, resolution of forces and balancing of moments leads to the
following equations for three forcesF 1 ,F 2 ,F 3 (Newtons) acting on one of the struts
in an aircraft wing.

F 1 −F 2 = 0

2 F 1 +F 2 − 2 F 3 = 20

F 2 −F 3 = 4

Find the forces by (i) Cramer’s rule (ii) matrix inversion.

3. The simple harmonic oscillator
   The position,x, of a particle performing simple harmonic motion about the origin at a
   timetcan be described through Newton’s laws of motion by the differential equation:

d^2 x

dt^2

+ω^2 x= 0

whereω^2 is a positive quantity. It is standard practice to introduce a new variable:

dx

dt

=y

to give thesecond order system:

dx

dt

=y

dy

dt

=−ω^2 x

which may be written, in matrix form as,

d

dt

x=

[

01

−ω^20

]

xx=

[

x

y

]

=Ax

For various reasons arising from the theory of differential equations (Chapter 15) we

now try to solve such systems by assuming solutions of the form

x=x 0 eλt

y=y 0 eλt

whereλis some constant to be determined, andx 0 ,y 0 will depend on the initial

conditions. Substituting in the equations gives the matrix form

[

−λ 1

−ω^2 −λ

][

x 0

y 0

]

= 0