10.1 SOME REAL EXAMPLES
Example 10.1 [MacPherson Strut]
Vib rations are a common phenomenon. The bouncing motion of an automobile serves
to illustrate the mathematics of vibrating motion. This
motion results from the displacement of a coiled suspension system. The
MacPherson strut, a popular suspension system, consists of a heavy
tubular strut that connects the wheel to the automobile frame. The wheel is
bolted or welded to the lower end of the strut as depicted in Figure
10.1.
A coiled spring surrounds the upper half of the strut, and a telescoping shock
absorber is set within the spring and strut as depicted in Figure
10.2.
In order to simplify the analysis , we make the following assumptions:
•
The strut is vertically aligned.
•
The automobile is not traveling but is subject to up-and-down motion.
•
The spring and the shock absorber have no mass; all the mass is in the load .
•
The base of the strut is fixed: the loading from the top causes compression .
The ODE for the motion of the block follows from Newton's second law ; we get
where is the mass of the load (typically 25% of the auto's mass), is the
displacement of the load from it's rest position at time is the
damping coefficient, is the spring constant, and is an external force (e.g, a
sequence of bumps in the road). In view of Example 1.5, we
see that
End
of Example 10.1
Example
10.2 [RLC Circuit]
An electric circuit consists of a set of basic circuit elements connected by
wires. A single loop or series circuit is illustrated in Figure 10.3.
According to Kirchoff 's voltage law (KVL), the charge across the capacitor is
modeled by the ODE
where 
is the charge on the capacator at time
is the inductance of the inductor
(also called a coil), 
is the resistance of the resistor,
is the capacitance of the capacitor, and 
is an external voltage source. Kirchoff's voltage law asserts that the sum of the voltage drops
across the components in a series circuit is zero. In particular, the voltage
drops across the components 
and
are
End
of Example 10.2
The physical parallels between the MacPherson strut and the RLC circuit are
depicted below in Figure 10.4.
Remark
10.3 [The Mechanical - Electrical Analogy]
We inter pret the parallels drawn in Figure 10.4. Electrical current
is the
time-derivative of charge 
Thus current is like
velocity - both are motions of some kind. And voltage is a kind of force - it is
what pushes the current through a resistor.
Current in an inductor is just like velocity of a mass - both keep going in the
absence of any voltage or force, respectively.
Shock absorbers are ana logous to resistors - the shock absorber resists motion
just as a resistor resists current, and both
dissipate, rather than store energy.
Springs also have a straightforward analogy in this scheme - they are like
capacitors. Running current into a capacitor,
building up voltage, is just like having a velocity compressing a spring,
building up force.
Thus it should not be too surprising to learn that the mechanical system and the
electrical circuit are modelled by the same
ODE
where 
and
are constants and
is a piecewise continuous function on some interval
We can simplify the ODE
somewhat by normalizing the coefficient of 
:
Finally we have the ODE
where 
and
Note that 
and
are constants. We will study linear second-order ODEs
with
time varying coefficients 
and
in a later lecture.
10.2 TERMINOLOGY
1. The ODE Eqn. (10.3),
is called a linear second-order ODE with constant
coefficients.
2. The function 
is called a forcing function (driving function or an input).
3. When 
is identically zero, Eqn. (10.3) is called homogeneous.
4. When 
isn't identically zero, Eqn. (10.3) is called nonhomogeneous.
The definition of a solution to a second order ODE follows from the general
formulation in Lecture 1. In view of the
piecewise continuous inputs permitted for linear first-order ODEs, we allow
solutions to be piecewise differentiable as well.
Definition
10.4 [Solution]
Assume f is piecewise continuous on R and 
and
are constants. A solution to Eqn.
(10.3)
is a function 
that is defined on some interval
so that when
is substituted for
x in Eqn . (10.3), we get an identity
in 
for all 
Like the case for linear first-order ODEs, we will develop a formula for the
general solution to Eqn. (10.3). This construction
is sufficient evidence for the existence of solutions and eliminates the need
for an existence theorem as was needed for
(nonlinear) first-order ODEs in Lecture 8. The uniqueness of a solution to an
IVP for Eqn. (10.3) will also follow from our
formula for a solution.
If f is continuous throughout
or has only jump discontinuities on
, then the
interval of definition 
that is, the solution 
is defined at all
10.3 THE LINEAR HOMOGENEOUS ODE
The rest of this lecture is devoted to the analysis of the solutions to a linear
homogeneous second-order ODE with constant
coefficicnts
In this case solutions are defined for all
A Search for Solutions
The first order linear homogeneous ODE
(with constant has an exponential
solution 
Indeed, any
constant multiple 
is also a solution. Aside from the fact that this solution can
be derived by simple methods of calculus ,
is reasonable solution based on the following observation. Rewrite the ODE as
This equation indicates
that the derivative of the solution is just a constant multiple of that
solution. The only such function of which we are aware
that has such a property is the exponential function.
According to Definition 10.4 of a solution, any solution
to Eqn. (10.4) must be
such that the linear combination
is zero on 
An exponential function of the form
should work as all higher
derivatives of 
are just multiples of 
we can attempt to find out what value(s) of
force
to be zero for all 
in some interval
Factor out the common term
so that
As is never zero for any value of it follows that
will be a solution to
for all
provided 
satisfies the quadratic equation 
Example
10.5
Determine all solutions of the form 
to the ODE
Solution: Set 
Then
and 
is to be a solution, then substitute
into the ODE to
obtain
Factor to get
As 
is never zero, we must have
Thus 
Set
Thus we have TWO solutions to the ODE, namely
End
of Example 10.5
The linear second-order homogeneous ODE
is called linear for good reason. It says that a linear combination of two
solutions is itself a solution.
