And now for something completely different…
The equations of motion for linear vibrating systems are well known and widely used in both mechanical and electrical devices. However, when students are introduced to these, they are frequently presented with solutions which are either essentially underived or inadequately so.
This brief presentation will attempt to address this deficiency and hopefully show the derivation of the equation of motion for an undamped oscillating system in a more rigourous way.
Consider a simple spring/mass system without a forcing function. The equation of motion can be expressed as
where x(t) is displacement as a function of time, m is the mass of the system, and k is the spring constant. The negative sign on the right hand side of the equation is not an accident, as the spring force always opposes the motion of the mass, and is the result of using a mechanical engineers’ “free body diagram” method to develop the equation.
Solutions to this equation generally run in two forms. The first is a sum of sines and cosines:
But it’s more common to see it in the form of
The latter is simpler and easier to apply; however, it is seldom derived as much as assumed. So how can it be obtained from the original equation?
Let us begin by considering the original differential equation. With its constant coefficients, the most straightforward solution would be a solution where the derivative (and we, of course, would derive it twice) would be itself. This is the case where the function is exponential, so let us assume the equation to be in the form of
(I had an interesting fluid mechanics/heat transfer teacher who would say about this step that “you just write the answer down,” which we as his students found exasperating, but this method minimises that.)
Substituting this into the original equation and diving out the identical exponentials yields
Solving for α yields
The right hand term is the natural frequency of the system, more generally expressed as a real number:
Thus for simplicity the solution can be written as
At this point it is not clear which of these two solutions is correct, so let us write the general solution as
Because of the complex exponential definition of sines and cosines, we see the beginning of a solution in simply one or the other, but at this point the coefficients are in the way.
These coefficients are determined from the initial conditions. Let us consider these at t=0:
Substituting these into our assumed general solution yields
The coefficients then solve to
It is noteworthy that the two coefficients are complex conjugates of each other.
Since the general solution is written in exponential form, it makes sense that, if the coefficients are to be removed so we can enable a direct solution to a sine or cosine, they too should be in exponential form. Converting the two coefficients to polar form yields
Substituting these coefficients into the general solution, we have
Factoring out the radical and recognising that the arctangent is an odd function,
The quantity in brackets is the complex exponential definition of the cosine, since the two exponents are negatives of each other. The solution can thus be written as
If we define
the solution is
which can be rewritten in a number of ways.
If the dampening is added, the problem can be solved in the same way, but the algebra is a little more complicated, and we will end up additionally with a real exponential (decay) in the final solution.
This derivation demonstrates the power of complex analysis as applied to differential equations even in a simple way.