Re: How Integral Calculus Models Reality
And now my light bulb came on. I think I finally understand you now. I draw a simple diagram and then model the forces acting upon uponit. Now that I understand the derivative and the integral that would be easier.
Exactly. Well done. By being able to mathematically describe (model) all the forces acting on the magnets, we can predict the performance of the system. In the case of your device, we can begin by only building a "1-DOF" analysis model. The main degree of freedom of motion that reactor's device will respond in is the VERTICAL DOF (up/down). So that is the 1 DOF we wish to write the force equations to describe. Let's call this up/down axis the "Z" axis. We can write the Newtonian equation of motion for this axis as:
Sum(Fz) = m*az (Summation of all Forces in the Z direction equals mass times the acceleration in the Z direction).
Sum(Fz) represents the sum of ALL forces acting in the Z direction. Meaning not only the force due to gravity of the magnet/object, but also the oscillating force created by the magnetic field of the magnets. If we can model all of these forces acting on the object, and we measure its mass we can solve the equation as:
az = Sum(Fz)/m (Equation computing the resulting acceleration on a body when subjected to Fz forces.)
From this equation we can build a time-integrator simulation of your device by simply integrating the az acceleration model over time, to predict its velocity (+ and -) along the Z axis. Then we integrate the velocity along the Z axis to predict the displacement (+ and -) along the Z axis from its rest position.
This basic process is exactly how all flight simulators do their job. We can accurately model how the aerodynamic forces act on the airplane (thrust, lift, drag, weight) over periods of time. Except we build these models for ALL SIX Degrees Of Freedom (Vertical, Lateral, Longitudinal, Pitch, Roll, and Yaw) of these aerospace vehicles. Our simulations are much larger, much more detailed, and as a result much more complex. But the very heart of all vehicle simulations are based on this modeling technique which uses powerful equations from physics. We literally use the math to model how a system will peform, before we ever really build it.
This is the basis for what we call model-based systems engineering. It is the most accurate and organized way to develop any complex control system, such as aircraft and spacecraft. And as you can see, it is all backed-up by mathematics that describe how reality really works. /ttiforum/images/graemlins/smile.gif
RMT