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View Full Version : Cushion Resiliency (Science....sort of)

Jal
09-13-2005, 02:37 PM
Here are some measurements of the before and after cushion impact cueball speeds, taken off of four of Dr. Dave's videos (kick shots straight into the cushion). The first number on each line is the incoming speed (mph), and the second is the after/before ratio.

fast polished follow: 12.8 .91
fast polished stun : 16.0 .73
fast polished draw : 12.8 .83

slow polished follow: 3.8 .69
slow polished stun : 4.7 .68
slow polished draw : 5.1 .81

fast follow : 11.6 .79
fast stun : 14.2 .69
fast draw : 11.6 .73

slow follow : 3.3 .63
slow stun : 6.1 .58
slow draw : 4.3 .60

Average fast speed ratio =.78 or .75 if the .91 ratio is thrown out.

Average slow speed ratio = .67

I expected the higher speeds to produce, if anything, a smaller ratio. I think the contrary result is a consequence of how I measured them and the fact that with the higher speed shots it only takes about 10-15 frames for the ball's diameter to pass a reference point (ruler placed over the screen). So an error of one frame in estimating when the edge of the ball is aligned with the edge of the ruler produces a considerable error in the calculation.

But I'm wondering if there is any reason to believe that the higher velocity shots do in fact retain more of their speed after the collision?

And, if not (if the ratio should be constant for any incoming velocity), what is it for reasonably live cushions? The above data suggests it's around .67, but I've seen .75 reported.

Jim

Bob_Jewett
09-13-2005, 04:06 PM
<blockquote><font class="small">Quote Jal:</font><hr> Here are some measurements of the before and after cushion impact cueball speeds, taken off of four of Dr. Dave's videos (kick shots straight into the cushion). The first number on each line is the incoming speed (mph), and the second is the after/before ratio. ... <hr /></blockquote>
One related observation is that rebound angle is only very weakly related to speed for a ball stunned into a cushion at an angle. I think this suggests that the speed ratio should be independent of speed.

In any of the measurements, did the cue ball get to smooth rolling after the cushion and before you measured its speed?

The observation on the Jacksonville tape, which looked at this problem only briefly, was that on normally worn cloth, the cue ball lost all its rotation while in the cushion for straight-on contact. I would expect this to mean that all the ratios should be the close to the same for draw/stun/follor for any particular speed.

Jal
09-13-2005, 05:54 PM
<blockquote><font class="small">Quote Bob_Jewett:</font><hr>...
In any of the measurements, did the cue ball get to smooth rolling after the cushion and before you measured its speed?<hr /></blockquote>

I tried to take the measurements close to the cushion to avoid the slow down resulting from this. But in reviewing the tapes, I see that for the slow shots with an unpolished ball, the ball did begin to develop some noticeable roll soon after leaving the cushion in the area where I measured it. So this would likely explain the smaller ratios for those shots. Appreciate the thought.

Jim

Bob_Jewett
09-14-2005, 07:35 AM
<blockquote><font class="small">Quote Jal:</font><hr> ...
I tried to take the measurements close to the cushion to avoid the slow down resulting from this. But in reviewing the tapes, I see that for the slow shots with an unpolished ball, the ball did begin to develop some noticeable roll soon after leaving the cushion in the area where I measured it... <hr /></blockquote>
If the ball leaves the cushion sliding, which seems to be normal for typical equipment, it will slow by a ratio of 5/7 as it acquires smooth rolling (and ignoring the minor effects of rolling friction slowing during that relatively short distance).

dr_dave
09-14-2005, 09:23 AM
Jim,

The effective coefficient of restitution (COR), the ratio of outgoing to incoming speed, can depend on rail impact height and rail and cloth conditions. I did some tests on my table a couple of years ago and found the average COR to be about 0.7. Here are some other COR numbers I've seen reported:

Coriolis: 0.6 for slow, 0.5 for fast
Marlow: 0.55

Interestingly, Coriolis found lower CORs at faster speeds ... but the rail materials and rail impact heights might have been a little different in the early 1800s. For source bibliography info, see Pool/Billiards Physics Resources (http://www.engr.colostate.edu/~dga/pool/physics.html).

Regards,
Dave

<blockquote><font class="small">Quote Jal:</font><hr> Here are some measurements of the before and after cushion impact cueball speeds, taken off of four of Dr. Dave's videos (kick shots straight into the cushion). The first number on each line is the incoming speed (mph), and the second is the after/before ratio.

fast polished follow: 12.8 .91
fast polished stun : 16.0 .73
fast polished draw : 12.8 .83

slow polished follow: 3.8 .69
slow polished stun : 4.7 .68
slow polished draw : 5.1 .81

fast follow : 11.6 .79
fast stun : 14.2 .69
fast draw : 11.6 .73

slow follow : 3.3 .63
slow stun : 6.1 .58
slow draw : 4.3 .60

Average fast speed ratio =.78 or .75 if the .91 ratio is thrown out.

Average slow speed ratio = .67

I expected the higher speeds to produce, if anything, a smaller ratio. I think the contrary result is a consequence of how I measured them and the fact that with the higher speed shots it only takes about 10-15 frames for the ball's diameter to pass a reference point (ruler placed over the screen). So an error of one frame in estimating when the edge of the ball is aligned with the edge of the ruler produces a considerable error in the calculation.

But I'm wondering if there is any reason to believe that the higher velocity shots do in fact retain more of their speed after the collision?

And, if not (if the ratio should be constant for any incoming velocity), what is it for reasonably live cushions? The above data suggests it's around .67, but I've seen .75 reported.

Jim <hr /></blockquote>

dr_dave
09-14-2005, 09:36 AM
<blockquote><font class="small">Quote Bob_Jewett:</font><hr> <blockquote><font class="small">Quote Jal:</font><hr> Here are some measurements of the before and after cushion impact cueball speeds, taken off of four of Dr. Dave's videos (kick shots straight into the cushion). The first number on each line is the incoming speed (mph), and the second is the after/before ratio. ... <hr /></blockquote>
One related observation is that rebound angle is only very weakly related to speed for a ball stunned into a cushion at an angle. I think this suggests that the speed ratio should be independent of speed.<hr /></blockquote>
I think the concept of coefficient of restitution (COR) becomes less clear if rebound angle is considered. With 0 rebound angle, the speed ratio gives the "normal" (or perpendicular) COR. With an angle into the rail, other factors come into play:
- cloth friction from sideways motion
- tangential (sideways) COR, AKA rail "throwback"
- changed normal COR due to sideways compression of the cushion.

I agree with you that the rebound angle is fairly insensitive to speed alone; although, in my tests, the angle typically goes a little "short" for faster shots and a little "long" for slower shot). However, this might be the result on many factors (listed above), each of which can change with speed. Although it seems like the changes in each effect due to speed tend to cancel each other out over a fairly wide range of speeds.

Regards,
Dave

Jal
09-14-2005, 11:35 PM
<blockquote><font class="small">Quote Bob_Jewett:</font><hr>]
If the ball leaves the cushion sliding, which seems to be normal for typical equipment, it will slow by a ratio of 5/7 as it acquires smooth rolling (and ignoring the minor effects of rolling friction slowing during that relatively short distance). <hr /></blockquote>

I redid the measurements and calculations to see if some of the data scatter could be reduced. The calculations now take into account the slowing of the ball (both inbound and outbound) as per your suggestion. Thanks.

Jim

Jal
09-15-2005, 12:55 AM
<blockquote><font class="small">Quote dr_dave:</font><hr> Jim,

The effective coefficient of restitution (COR), the ratio of outgoing to incoming speed, can depend on rail impact height and rail and cloth conditions. I did some tests on my table a couple of years ago and found the average COR to be about 0.7. Here are some other COR numbers I've seen reported:

Coriolis: 0.6 for slow, 0.5 for fast
Marlow: 0.55

Interestingly, Coriolis found lower CORs at faster speeds ... but the rail materials and rail impact heights might have been a little different in the early 1800s. For source bibliography info, see Pool/Billiards Physics Resources (http://www.engr.colostate.edu/~dga/pool/physics.html).<hr /></blockquote>
Hi Dr. Dave. Good to see you back and thanks for the resource link.

It's interesting that Marlow got 0.55 for the COR (in this day and age). Excessively high cushions? Also, that Coriolis found the COR to diminish with speed. If you consider the cushion to be a damped oscillator, shouldn't this be true? Yet the data seems to indicate, as Bob Jewett noted, that it ain't necessarily so.

I did the measurements and calculations again and the unpolished ball results do agree very well with what you obtained for your table (I hadn't read your post beforehand).

Instead of a ruler for a reference line, I used a thread taped to the screen to better judge when an edge was aligned with it. The computations also make the adjustments for the slight slowing down of the ball both during the measurement interval (one diameter of ball movement) and the distance to and from the rail during the inbound and outbound phases. For the unpolished ball they assume a coefficient of friction of 0.2. For the polished ball, I determined the coefficient to be about 0.13, from the amount of spin removed from the fast follow shot.

The first number is the incoming speed and the second the after/before speed ratio (COR). The asterisks indicate those shots where the leading/trailing edges were very visible (white) for all the measurements (as opposed to being a little indeterminite due to the dark stripe blending with the background).

fast polished follow: 12.7 0.84
fast polished stun : 15.3 0.71*
fast polished draw : 13.0 0.76

slow polished follow: 3.6 0.76*
slow polished stun : 4.3 0.69
slow polished draw : 5.0 0.79

fast unpolished follow: 11.6 0.76*
fast unpolished stun : 14.2 0.70
fast unpolished stun : 15.4 0.64
fast unpolished draw : 11.4 0.73*

slow unpolished follow: 3.0 0.74*
slow unpolished stun : 5.5 0.67*
slow unpolished draw : 3.8 0.69*

Ave speed ratio fast polished = .77
Ave speed ratio slow polished = .75
Ave speed ratio fast unpolished = .71
Ave speed ratio slow unpolished = .70

Average of all shots = .73
Average of all shots with the clearest view of edges (asterisks) = .72

Although the data scatter has been reduced, there seems to be a definite difference between the unpolished and polished ball shots. Perhaps the result of bed friction during impact?

Jim

dr_dave
09-15-2005, 07:51 AM
<blockquote><font class="small">Quote Jal:</font><hr>It's interesting that Marlow got 0.55 for the COR (in this day and age). Excessively high cushions?<hr /></blockquote>
I think you are right. In Marlow's book, he suggests that cushion impact heights should be at 7/10 of the ball diameter (the center of percussion), so a ball rolling into the cushion would tend to also roll away from the cushion (after rebound). Then he points out that manufacturers typically design the cushions and rails so the impact height is lower than 7/10 D (usually closer to 6/10 D). He suggests (in a footnote on pg. 175) that they do this to reduce "rail groove" wear on the cloth. He doesn't state what type of equipment he used or whether he assembled his own rail for testing purposes, so I don't know the value of his rail impact height.

<blockquote><font class="small">Quote Jal:</font><hr>Also, that Coriolis found the COR to diminish with speed. If you consider the cushion to be a damped oscillator, shouldn't this be true? Yet the data seems to indicate, as Bob Jewett noted, that it ain't necessarily so.<hr /></blockquote>My initial intuition would have me guess a lower COR at higher speeds, but I don't deny the experimental results. Honestly, nothing would surprise me here, because the physics of cushion collisions is very complex. The rubber material and geometry create lots of nonlinearities, and friction from the cloth (both on the cushion and the table surface if the ball is driven down) plays a large role. One possible explanation for the higher effective COR at higher speeds is that friction is generally lower at higher speeds, so there might be less damping from friction at the higher speeds.

<blockquote><font class="small">Quote Jal:</font><hr>I did the measurements and calculations again and the unpolished ball results do agree very well with what you obtained for your table<hr /></blockquote>
Thank you so much for putting in all of the effort to analyze the video clips. It is nice to have the numerical results to go along with the clips. The values you report are also consistent with values I have measured in other tests.

<blockquote><font class="small">Quote Jal:</font><hr>The first number is the incoming speed and the second the after/before speed ratio (COR). The asterisks indicate those shots where the leading/trailing edges were very visible (white) for all the measurements (as opposed to being a little indeterminite due to the dark stripe blending with the background).

fast polished follow: 12.7 0.84
fast polished stun : 15.3 0.71*
fast polished draw : 13.0 0.76

slow polished follow: 3.6 0.76*
slow polished stun : 4.3 0.69
slow polished draw : 5.0 0.79

fast unpolished follow: 11.6 0.76*
fast unpolished stun : 14.2 0.70
fast unpolished stun : 15.4 0.64
fast unpolished draw : 11.4 0.73*

slow unpolished follow: 3.0 0.74*
slow unpolished stun : 5.5 0.67*
slow unpolished draw : 3.8 0.69*

Ave speed ratio fast polished = .77
Ave speed ratio slow polished = .75
Ave speed ratio fast unpolished = .71
Ave speed ratio slow unpolished = .70

Average of all shots = .73
Average of all shots with the clearest view of edges (asterisks) = .72

Although the data scatter has been reduced, there seems to be a definite difference between the unpolished and polished ball shots. Perhaps the result of bed friction during impact?<hr /></blockquote>Sounds plausible to me.

Again, thank you for your good work,
Dave

Jal
10-30-2005, 12:49 PM
<blockquote><font class="small">Quote Jal:</font><hr>
Ave speed ratio fast polished = .77
Ave speed ratio slow polished = .75
Ave speed ratio fast unpolished = .71
Ave speed ratio slow unpolished = .70
....
Although the data scatter has been reduced, there seems to be a definite difference between the unpolished and polished ball shots. Perhaps the result of bed friction during impact?<hr /></blockquote>Just a footnote.

I've tried taking into account the slowdown that occurs during impact due to bed friction. Using the average speed ratio of .70 for the unpolished ball and .75 for the polished ball, and using 0.2 and 0.13 for the respective coefficients of friction, produced the following values for the cushion's coefficient of restitution.

Unpolished: e=.80
Polished : e=.82

If nothing else they're consistent. This is the speed ratio in the horizontal direction (parallel to the bed) when the effects of bed friction are removed.

Jim

wolfdancer
10-30-2005, 01:09 PM
If I may quote from Ernest Jones...1952, with pool substituted for golf:
"I have no quarrel with those who delight in analyzing....many of us are curious to take a watch apart to see what makes it tick....the avg player must be allowed to theorize to some extent. It is a necessary concession to him as a thinking animal. On the other hand if he does not recognize that stroking the ball is his main business, and theory is his recreation....he may give the game up."

wolfdancer
10-30-2005, 01:13 PM
Say, what would be the coefficient of restitution, if i lost \$100 to a hustler? would I need the full c-note back, plus table time?....or should I also inflict bodily harm? Well, I'd have my big brother take care of the menial details

Jal
10-30-2005, 03:20 PM
<blockquote><font class="small">Quote wolfdancer:</font><hr> If I may quote from Ernest Jones...1952, with pool substituted for golf:
"I have no quarrel with those who delight in analyzing....many of us are curious to take a watch apart to see what makes it tick....the avg player must be allowed to theorize to some extent. It is a necessary concession to him as a thinking animal. On the other hand if he does not recognize that stroking the ball is his main business, and theory is his recreation....he may give the game up."
<hr /></blockquote>e=mc^2 seemed useless at the time....and would that it were.

Who knows where e=.8 will lead us?

Jim

dr_dave
10-31-2005, 08:12 AM
<blockquote><font class="small">Quote wolfdancer:</font><hr> Say, what would be the coefficient of restitution, if i lost \$100 to a hustler? would I need the full c-note back, plus table time?....or should I also inflict bodily harm? Well, I'd have my big brother take care of the menial details <hr /></blockquote>
Cute /ccboard/images/graemlins/grin.gif

To answer your question, if you lost \$100 to a hustler, the COR is 0.

Dave