# Applications of overdamping?

1. Jun 6, 2014

### bcrowell

Staff Emeritus
It's easy to think of mechanical devices that are underdamped (pendulum, guitar string) or critically damped (automatic door closers, various control systems such as cruise control). But what is a good, simple, pedagogical example of a practical mechanical device that is overdamped, and for which the overdamped behavior is desirable? In a control system, you typically want to get to the equilibrium state as fast as possible, so you would want to tune the damping to be critical.

I'm especially interested in simple examples for which the free rather than the driven response is of interest. (I think there are things called tuned dampers for car engines and skyscrapers that one would like to be overdamped if possible, but I don't know if they are overdamped in practice, and these are examples where you're interested in the driven response.)

2. Jun 6, 2014

### Staff: Mentor

Handicap-access door closer mechanisms

Push-button water faucet shut-off valves (like you see in public restrooms)

Recoil return dampers for artillery pieces (the actual recoil absorption is probably critically damped, I would guess)

(basically anything that you want to return slowly to its original position...)

3. Jun 6, 2014

### bcrowell

Staff Emeritus
I could be wrong, but I believe these are all designed to be *critically* damped. I've seen both the door closer and the gun mechanisms described as being tuned for critical damping.

4. Jun 6, 2014

### Staff: Mentor

I picture each of them as overdamped because the motion looks almost linear (in highly over-damped systems). Critically damped systems have motion that looks almost exponential. BTW, the gun recoil recovery motion that I'm referring to is after the initial recoil is absorbed, and the piece is returning to the forward firing position. I'll see if I can scare up some videos...

http://www.efunda.com/formulae/vibrations/sdof_images/SDOF_OverDamped_Response.gif

5. Jun 6, 2014

### AlephZero

You may also want to avoid any overshoot, even if the initial conditions have the system moving in the "wrong" direction. A moderate amount of overdamping might be desirable to achieve that. A textbook analysis of the step response of a system probably assumes the initial velocity is zero.

Your question seems to be about single degree of freedom systems, and tuned dampers are basically 2 DOF systems.

They work on a different principle from the simple idea of "damping out an oscillation". The basic function is to transfer energy from where it can cause damage (e.g. torsional vibration of the engine crankshaft or motion of the skyscraper) to somewhere else where it is harmless and can then be dissipated (e.g. an oscillating metal ring for the crankshaft damper, or a massive pendulum for the skyscraper.). The "energy absorber" is not likely to be overdamped, because it has to be able to move around to gain kinetic energy. If it was heavily overdamped, it would behave more like an additional mass rigidly connected to the crank or skyscraper, which would change the system's vibration frequency a bit, but would not take much energy out of the system.

Remember that real dampers are usually nonlinear devices, so the notion of "critical damping" is rather idealized. For example a friction damper generates a force that is approximately constant, not a force proportional to velocity. Hysteretic damping (strain energy in the material converted to heat) removes a constant proportion of the strain energy in each cycle of vibration, independent of the vibration frequency. A damping force caused by "air resistance" will probably be proportional to velocity squared. A damper that forces fluid through an orifice isn't linear either.

For a real-world lightly damped system, the nonlinear behavior of the damper doesn't have much effect on the approximation that the motion is harmonic, and for practical purposes you can model the system with a linear damper that takes out the same amount of energy per cycle as the real-world nonlinear one. But that linear approximation breaks down as the damping level increases.

The best real-world examples of linear damping are probably electrical, not mechanical. Ideal resistors are very good approximations to real ones.

Last edited: Jun 6, 2014
6. Jun 7, 2014

### AlephZero

That was not very carefully worded (and might even be interpreted wrongly) so to spell out the math:

Consider the critically damped system $\ddot x + 2 \dot x + x = 0$ with $x(0) = 1$.

The general solution is $x(t) = (1 + at)e^{-t}$ where $a$ depends on $\dot x(0)$.

If $a >= 0$, $x$ is always positive, i.e. the response does not overshoot the equilibrium position $x = 0$.

But if $a < 0$, the response passes through 0 when $t = -1/a$, overshoots, and then returns towards 0.

For the "door closer", that corresponds to what happens if you slam the door shut instead of just letting it go.

If the system is overdamped, overshooting can still happen but it needs a bigger (negative) initial velocity to cause it.

In an application like computer controlled machining, or landing a plane by autopilot, overshooting the desired response could have worse consequences than slamming a door shut!

7. Jun 8, 2014

### bcrowell

Staff Emeritus
Aha, excellent examples!