Previously I showed that the rocking suitcase demonstrates something tolerably close to simple harmonic motion (SHM) even though the forcing acting on it is a step-function that changes sign at zero, rather than a linear ramp.
The frequency of the motion is amplitude-dependent, with the relation:
ω ~ sqrt(12*g/(5*L*A0))
where A0 is the amplitude in radians, L is the width of the wheels and the suitcase (assumed to be the same) and g is a well-known constant.
For a rotational simple harmonic oscillator the characteristic frequency is:
ω = sqrt(k/I)
where k is the torsion constant (the restoring force per angular displacement) and I is the moment of inertia about the centre of rotation. We can use this to get an “effective k” from measurement or simulation of the rocking suitcase. Otherwise, from the simple theory, the effective k is:
k = m*g*L/A0
This is important because damping is k-dependent, and in particular, critical damping is k-dependent. Knowing the effective k is going to let us figure out what the optimal damping constant is.
For SHM, there is a state known as “critical damping” in which the system returns to equilibrium as rapidly as possible. Less than critical damping and thing oscillate for a while. More than critical damping and things come back to equilibrium on a slower, longer curve.
The condition for critical damping is:
c/(2sqrt(k*I)) = 1
where c is the damping force coefficient and k and I are defined as above. This is called the “damping ratio” and usually given the symbol ζ (Greek letter zeta). There is a rich convention of notation in physics, with Greek letters being used for angular qualities (α for angles, ω for angular frequencies) and particular concepts within specific fields. It is not uncommon to see the letter take over as the name of the concept, as in the “plasma β” for the charged-fluid analog of pressure, for example.
Based on the simple theory, we get a c-value of:
c = 2*sqrt(12*g/(5*L*A0))
which gives us damping coefficients around 21 N*s/radian for half a radian motion. Plugging this into the equations of motion, we get the following result:
Once again the value of SHM as a model for things that aren’t necessarily very close to the underlying assumptions comes to the fore. I’ve based the damping coefficient on a ridiculously simple approximation to the actual case, and yet it results in something that is remarkably close to critically damped motion, with a tiny over-shoot. The weirdly linear shape of the ramp is presumably due to the constant force law interacting with the damping force.
The question remains, then: how does one implement, physically, the damping described by the equation above? This is where physics shades into engineering. Physics and engineering involve very different kinds of creativity. The former requires that we figure out what laws are operative (sometimes inventing new laws in the process) and sometimes invent new ways of applying them. Physics in the “normal science” case tells us what will happen if we create configuration X.
Engineering, in contrast, is about coming up with configurations such that they are described by the equations physics gives us. What physical system will result in a damping coefficient that scales like the reciprocal of the square root of the amplitude of the motion? I have a couple of ideas, but they are more about intuition than anything else, and will no-doubt require some empirical testing to validate, disprove, or improve.
The important thing is that I have a fair idea of the damping required, and a good idea of the scaling law that describes the damping. How significant the latter is remains an open question: maybe a constant, small damping force coefficient will be enough to prevent things from ever getting large. Maybe there is a characteristic scale to the driving impulses (the size of the bumps wheeled suitcases hit and the speed they are traveling) that is such that a constant damping force coefficient will kill the motion in that regime before it ever has a chance to get large. These are not questions that can be answered without actually gathering some data.
I’m pretty sure I have a board or two with an accelerometer on it. Maybe it’s time to instrument a suitcase and visit the airport…