Zener showed (C. Zener, "Intrinsic Inelasticity of Large Plates," Physical Review, vol. 59, 1941, pp. 669-673) that a thin flat plate reacts to an impact by having a velocity at the impact point that is proportional to the applied force. This is the essence of a shock absorber. In the flat plate, however, the energy of the impact is still in the plate, in the form of flexural waves. This energy gets dissipated by internal friction of the plate material, which heats the plate a little. Unless the plate is very, very large this energy will be reflected by the plate supports and return to the contact point, causing a bounce. We actually depend on a very slight bounce to give the plate material a few milliseconds to dissipate the energy of impact before the impacting object touches the plate again. This bounce and secondary impact are so slight as to be unnoticeable to the naked eye.
You can try it out by dropping a marble onto a inverted cookie sheet. The sheet thickness has to be less than about a fifth of the ball diameter and the width and length have to be much larger than the ball diameter. While the visual effect of such an impact is quite startling, the flat plate leaves much to be desired as an engineering device. That's why I made some improvements on it and patented it. Hopefully, someone who is tired of having things bounce around on an assembly line will be happy to pay me for the privilege of using the patented design features. I have no intention of manufacturing and selling such devices.
If the flat plate acts like a shock absorber, why not just use a shock absorber to get a very low coefficient of restitution? The first reason is that a shock absorber has a limited life, but a properly designed flat plate should last forever. A second reason is that shock absorbers tend to be messy, especially if a seal fails and oil leaks out. A flat plate obviously has moving parts, but none that rub together, so the device should very clean.
The "Mechanical Energy Absorber" (U.S. Patent Number 6,006,874) patent addresses several of the shortcomings of Zener's flat plate. Among these shortcomings are limited allowable plate deflection (only half the plate thickness), material damage at the contact point, difficulty in analyzing the plate response because of the nonlinear spring, the large size of plate required, and a lack of any control over the mass and velocity of the impacting body.
You can see the actual patent. If your web browser has trouble displaying the patent, you can look at this PDF version.
When a flat plate deflects more than about half the material thickness, the membrane tension stresses become significant and the bending stresses less so. The desirable plate response is due to the bending of the plate material, so large membrane stresses must be avoided. The source of this problem is pretty easy to see: the shortest distance between two points is a straight line and the sinusoidal shape of a flexural wave is not a straight line. Around any deflected region in the plate, you can draw paths where the undisturbed material would have to expand laterally in order to accommodate the wavy material inside. This undisturbed material is extremely rigid, resulting in very large tensile stresses for very small changes in length. The internal damping of the material, which is what really dissipates the impact energy, depends on the bending stress squared, so we want to allow large bending stresses. One way (covered in the patent) to allow large bending stresses in a flat plate without being limited by membrane stresses is to put radial slits in the plate. The slits break up the surrounding material and limit the tensile and compressive stresses that can be set up.
The radial slits in the thin flat plate shown in the video on the home page are what allowed us to use a .012"x24"x24" sheet of steel to stop a very bouncy ball without any perceptible bounce.
Where the impacting ball meets the plate, very high contact stresses can be set up in both the ball material and the plate material. The resulting deformations act as the nonlinear spring in Zener's analysis, but they can also cause material damage, such as peening, pitting, and spalling. This not only limits the life of the device, but also is a source of debris.
The nonlinear spring in Zener's analysis results in a differential equation that has to be evaluated numerically. A linear spring not only results in a differential equation that has a closed-form solution, but also makes the spring rate independent of the geometry of the impacting object.
Of course, the device can't change the mass and velocity of the impacting object, but it can change the effect of the impact on the flat plate. By using a lever between the impacting object and the flat plate, the plate "sees" a different mass and velocity with the same kinetic energy. The lever can be on either side of the linear spring, or can actually act as the spring.
So, where could this device be useful? How about where things are manufactured and they get popped out of the equipment and bounce around? Or as a shock absorber in the food or drug industries where leaking hydraulic oil would be a bad idea? Or as a shock absorber that would have an infinite life? One application that seems interesting to me, maybe just because I don't know anything about it, would be to reverse the flat plate by building it into a hammer that would act "dead", as if it were made of lead. I haven't been too interested in military or space applications because the federal government had an interest in this patent (I was indirectly working for them when this all started) so they wouldn't have had to pay to use it. It would be nice to hear that a Mars or Lunar rover used some features of this patent for their shock absorbers. My grandkids would love that.
Another possible application would be in a speed controller, called a runaway escapement, used in some timers. The time delay depends on a gear getting a part rotating, then stopping it against another part of the gear. Any bouncing in the impacts reduces the effect and causes variation in the timing.
I have put the equations for a flat plate device using a linear spring into a PDF document. These results are very similar to those in Zener's paper, but have a closed-form solution, making them much easier to use.
An improvement on Zener's flat plate that was also patented is using a plate made up of many thin layers instead of a single thicker layer. The advantage to this design is a reduction in the lateral dimensions of the plate, perhaps making the device easier to package. As near as I can tell, the total mass of the plate is unchanged. A small PDF document discusses the required changes in the equations.
I tried making a flat plate per Zener's article that would stop a generic version of a Super Ball©, using a plate material about as soft as the ball. I succeeded, but you wouldn't want to make an engineering device that way.