A frequently discussed example in differential equations and in analog computers is a mass suspended from a spring where there is viscous damping. There are three forces acting on such a mass. The gravitational force is given by
where g is the gravitational constant. The spring exerts a force to return the mass to a neutral state which is proportional to the distance away from the neutral position
where y=0 is the neutral postion for the spring. While the mass is in motion, there is a force acting against that motion and proportional to the velocity:
From Newton's laws of motion, we know that
which, in this case, gives us
Solving for the highest order derivative, we have
Once we express our differential equations in this form, we now create a circuit of computing elements that solves the equations. We assume the existance of a voltage (or a set of voltages that sum to one) proportional to the highest order derivative. Once we have that, we can easily create a voltage proportional to the negative of the next lower derivative by passing the higher order one through an integrator. We chain a number of integrators together to create the variable itself. For this second-order equation, we need two integrators. Now that we have each of the derivatives, we construct the weighted sum that is our highest order deravitive and feed that into the input of the first integrator.
The spring and mass equation gives a circuit shown in Figure 9.
Figure 9: Solution to the Spring and Mass Problem
Frequently, the constants of a problem would yeild difficulties if we just applied this procedure for solution. For example, the various values might experience fluctuations that are so slow that we waste time simulating them in real time. Or, more serious, the fluctuations might be so fast that the op-amp circuits have problems operating correctly or that our output devices can't keep up. Similarly, the range of values for the various points in a circuit might go out of range for our devices or they may alway be small increasing the susceptability to error. In practice, we use several rules-of-thumb to estimate behaviors such as these and create scaling factors that make the range of values usable. The effects of such scaling are most easily seen by way of a substitution of the troublesome variable by a constant times a new variable. These substitutions are done prior to the development of the circuit.