Second Order Differential Equation Example Request

Question

I am looking for some non-complicated second order differential equations to illustrate certain techniques for control engineering. It doesn’t matter if the differential equations are linear or non-linear; also, simplified assumptions are fine. I’d like to have a wider pool of examples.

I am aware of the standard example of the spring pendulum, and also the rolling motion of a plane simplified defined by: $$M(t) = k \ \delta_A(t)$$ and $$M(t) = I_{xx}\ \ddot \Phi(t)$$ with $M(t)$ being the moment, $\Phi(t)$ being the roll angle acceleration and $I_{xx}$ being the inertia torque.

Question: Can someone direct me to more realistic examples of this nature, or is there a reference or text containing a list of this sort somewhere?

Answer 1

Some examples from Erwin Kreyszig's Advanced Engineering Mathematics 7th ed. (John Wiley & Sons, Inc., 1993):

Falling stone
$\frac{\mathrm{d}^2y}{\mathrm{d}t^2}=g$
where $y$ is the position of the stone and $g$ is the gravitational acceleration

Hanging cable
$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=k\sqrt{1+\frac{\mathrm{d}y}{\mathrm{d}x}^2}$
where $y$ is the curve of an inextensible flexible homogeneous cable hanging between two fixed points and $k$ is a constant depending on weight

Vibrating mass on a spring
$m\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+ky=0$
where $y$ is the displacement of the body from the static equilibrium position, $m$ is the mass, and $k$ is the spring modulus

Current in an RLC circuit
$L\frac{\mathrm{d}^2i}{\mathrm{d}t^2}+R\frac{\mathrm{d}i}{\mathrm{d}t}+\frac{1}{C}i=\frac{\mathrm{d}v}{\mathrm{d}t}$
where $i$ is the current, $R$ is the resistance, $L$ is the inductance, $C$ is the capacitance, and $v$ is the voltage of the voltage source

Pendulum
$mL\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}+mg\sin\theta=0$
where $m$ is the mass of the bob, $g$ is the gravitational acceleration, $\theta$ is the angular displacement measured counterclockwise from the equilibrium position and $L$ is the length of the (massless) rod

Answer 2

I think Joel gave you the standard list.

Caveat one: I am used to control systems courses (well OK...the single college survey course I took) being essentially DE reviews. 2nd order ODE is huge but also some LaPlace transforms, etc. Maybe one different thing is doing some modeling and learning to draw those little block diagrams. But still very mathematical and not even much real modeling/applications. But perhaps text I had was bad this way (and if so, it was bad on purpose...they felt covering the math was more global and helpful in the future...I kinda disagreed since it seemed like such a repeat of DEs!) In contrast, taking a practical mechE engineer/contractor course from Landys and Gyr, it was much more about listing numbers of control points, types of controllers (analog/digital, PID, etc.) More about cataloguing I/O points so you can buy something and wire it. There probably should be (probably is) something between those two extremes...but I don't know.

Caveat two: I don't think your 'has to be non complicated' and 'fine if it is nonlinear' are self consistent.

Answer: Some standard application areas (sorry I don't have all the details...would think they are in common texts):

1. Building HVAC or chemical process controls. Your students will deal with in workplace, mechE/chemE. These are huge real work areas so anything you can do here, even with simplified examples will resonate.

2. Military systems: the dampened spring shock absorber of aircraft landing gear (very hard/important problem for example for heavy jet landing on an aircraft carrier). Also swinging a gun from position A to position B (you want it to happen fast, but you also have to limit the final oscillations).

3. Everyday life: A final interesting example is how hydraulic door closing mechanisms work--had a great chat with a smart janitor about how these work--there is a main spring, main damper and then even a little adjustment that gives it the "final push". You might even have one on the classroom door and be able to do some demonstration or messing with it...kids love stuff like that.