Systems of First Order ODEs
Eigenvalues and eigenvectors:
Repeated eigenvalues
Repeated Eigenvalues
In the ODE if the nby nmatrix is symmetric then it will
always have nindependent eigenvectors, no matter the multiplicity of
its eigenvalues. The fundamental solutions to the ODE will have the
form
xAx =
A
Repeated Eigenvalues (2)
If is not symmetric and it has a repeated eigenvalue then it may not
have nindependent eigenvalues and the fundamental solutions can be
more complicated. Three examples:
Example: (SP 8.6 in text)
A
Repeated Eigenvalues (3)
The associated fundamental solutions have the form
where
=−
0)(
IrA
Repeated Eigenvalues (4)
Example 2:
The matrix
111
Repeated Eigenvalues (5)
)(
)1(
rt
etx
=
The associated fundamental solutions have the form
where
=−
0)(
IrA
Systems of First Order ODEs
Coordinate Systems:
Road Vehicle Suspension System
Dynamics Example
Objective
To demonstrate by example that a fortuitous choice of dependent
We can think of this choice as selection of a coordinate system.
Road Vehicle Suspension System Model
Model abstractions:
Suspension springs can be coils (as shown), leaf springs,
Model simplifications:
Tires, wheels and undercarriage have negligible weight
Vehicle Suspension System Model (2)
Vehicle Suspension System Model (3)
2
y
1
y