Dear sammy,
I think your second post is great, and bears more scrutiny and consideration. I have often wondered how the driver-head gear effect related to the hoseling of the shaft tip and how this is probably different than the one-piece shaft hoseling of a putter. I believe there is much more here to explore.
This is what I was writing in response to putter face rising and impact dynamics before your second post about the principal vertical axis:
I like the approach you have here, but want a wee bit more detail if possible.
RISING OF PUTTER FACE IMPACT
According to calculations, 1 degree forward stroke raises the sole of the putter 0.0082245" off the surface where it began. At
this Circle Geometry online calculator, calculate the "segment height" of the circle using known radius (54") and angle in degrees of the stroke forward of center (1/2 the "central angle"). For example, to calculate the rise of the putter head over a 1-degree forward stroke, use radius=54" and central angle=2 degrees. This means the impact point on the face is 0.84" (height of equator) minus 0.0082245" (rise of sole lowers impact point on face), assuming the sole matched the bottom of the ball to begin with and the golfer's stroke is only pendular without extra lifting or lowering over this stroke region.
LOWERING OF BALL IMPACT
To look at this in terms of impact point on the back of the ball, the radius=0.84" and 1/2 the central angle is 1/2 of the design static loft plus the added dynamic loft. for a 3-degree putter with 1-degree added dynamic loft, the "presentation loft" is 4 degrees. The vertical length where a 4-degree loft hits below the equator is, then, calculated using radius=0.84" and central angle= 8 degrees. The impact point is then 0.0020462" lower than the equator.
Your example is a 3-degree putter with NO added dynamic loft. The calculation is then radius=0.84" and central angle= 6 degrees, so the impact point is 0.0011512" lower than the equator. You calculate this as "the initial contact point on the golf ball by a 3º loft putter is only 0.044" below the equator". So my calculation favors your point even more than your calculation, by a factor of 40 times.
FORWARD PRESS DELOFTING PUTTER FACE
The same website calculates how many inches moving the top of the putter handle parallel to the putter face aim line towards the target will deloft the face. For this, you want 1/2 the chord as the lateral movement of the top of the putter, with the radius the same as the length of the putter. So a 2-inch "forward press" (chord=4") delofts a 35-inch putter 1/2 the central angle, which is 1/2 of 6.5517 degrees or 3.27585 degrees.
From this point forward, I don't agree with the details. The exact point of impact is initially REALLY only a precise point. During impact, the putter face and ball cover both "smush" together, with the putter face giving very little and the cover of the ball giving most, but both actually give more than none. Both the putter face metal and the ball cover material have independent coefficients of restitution. Once impact begins, it is true, the putter face smushes the cover material on the ball flatter. How flat depends upon how deep from the spheroid surface the cover gets compressed towards the center of the ball. That in turn depends upon the relative masses and momenta of the ball and putter face at impact (or, roughly, the putter head velocity thru the ball center at impact, since the ball is stationary.) And to some extent this deepness depends upon the geometry of the dimple(s) involved in the impact and the geometry of the putter face (flat vs ridged).
Typically, a ball over 360 degrees around the equator has 24 dimples or so, meaning that one dimple occupies 360/24 of the circumference of the ball for the dimple diameter, which is 15 degrees of a ball with a diameter of 1.68" (4.267 cm) and a circumference of 5.2752" (13.4078 cm), so the typical dimple has a diameter of an "arc" of the sphere surface that is 15 degrees and 5.2752"/24 = 0.2198" (0.5586583 cm or 5.586583 mm). If the "ball" is 1.68" in diameter, then the diameter extends from the center of the sphere to the TOP of the dimple. Now we need a reasonable idea of the depth of the typical dimple from top edge of dimple to floor of dimple, as our "arc" is measured across the top edge of the dimple, not the floor.
Wikipedia reports: "Golf ball dimple size is from 0.14 to 0.17 inches." That is 3.556 mm to 4.318 mm. Yahoo! Answers says: "The diameter of one of them [a dimple] is between 2.0 and 2.7 mm." The above 1/24th the circumference of the ball includes some margin of surface between dimples, but the typical dimple is closer to 5 mm in diameter.
This recent patent for golf ball dimples has dimples with the following geometrical profile:
United States Patent US7163472. According to this design, the dimple is 0.025" deep (0.635 mm) with a radius of 0.05" (diameter= 1/10th inch, 0.10", 2.54 mm).
According to
US Patent 4681323, "The maximum diameter of the dimple [in this patent] can be within the range of 2 to 4 mm and the maximum depth of the dimple can be within the range of 0.1 to 0.4 mm." And: "[0042]The cover has a thickness of equal to or greater than 0.3 mm, and particularly equal to or greater than 0.4 mm. The cover has a thickness of equal to or less than 2.5 mm, and particularly equal to or less than 2.2 mm. The cover has a specific gravity of equal to or greater than 0.90, and particularly equal to or greater than 0.95. The cover has a specific gravity of equal to or less than 1.10, and particularly equal to or less than 1.05. The cover may be composed of two or more layers.
[0043]FIG. 2 shows an enlarged front view illustrating the golf ball shown in FIG. 1, and FIG. 3 shows a plan view of the same. As is clear from FIG. 2 and FIG. 3, all dimples have a circular plane shape. In FIG. 3, types of the dimples are indicated by reference signs A to D in one unit provided when the surface of the golf ball 2 is comparted into 20 equivalent units. This golf ball has dimples A having a diameter of 4.40 mm, dimples B having a diameter of 4.00 mm, dimples C having a diameter of 3.85 mm, and dimples D having a diameter of 3.80 mm. The number of the dimples A is 100; the number of the dimples B is 112; the number of the dimples C is 60; and the number of the dimples D is 60. Total number of the dimples is 332.
....
[0069]A rubber composition was obtained by kneading 100 parts by weight of polybutadiene (trade name "BR-18", available from JSR Corporation), 30 parts by weight of zinc diacrylate, 6 parts of zinc oxide, 10 parts by weight of barium sulfate, 0.5 parts by weight of diphenyl disulfide and 0.5 part by weight of dicumyl peroxide. This rubber composition was placed into a mold having upper and lower mold half each having a hemispherical cavity, and heated at 170° C. for 18 minutes to obtain a core having a diameter of 40.0 mm. On the other hand, 50 parts by weight of an ionomer resin (available from Du Pont-MITSUI POLYCHEMICALS Co., Ltd.; trade name "Himilan 1605"), 50 parts by weight of other ionomer resin (available from Du Pont-MITSUI POLYCHEMICALS Co., Ltd.; trade name "Himilan 1706") and 3 parts by weight of titanium dioxide were kneaded to obtain a resin composition. The aforementioned core was placed into a mold having numerous protrusions on the inside face, followed by injection of the aforementioned resin composition around the spherical body by injection molding to form a cover having a thickness of 1.35 mm. Numerous dimples having a shape inverted from the shape of the protrusion were formed on the cover. A clear paint including a two-part liquid curable polyurethane as a base was applied on this cover to give a golf ball of Example 1 having a diameter of 42.7 mm and a weight of about 45.4 g. The compression of this golf ball measured with a tester from Atti Engineering Co. Ltd., was about 85. This golf ball has a dimple pattern shown in FIG. 2 and FIG. 3. The dimples of this golf ball have a cross-sectional shape shown in FIG. 4 and FIG. 5. Details of specifications of the dimples are presented in Table 1 below."
In general, the depth of a dimple is on the order of 1/10th the dimple diameter.
US Patent 4142727.
BALL COMPRESSION DURING PUTTING
http://www.golfball-guide.de/knowledge.htm:
Definition: Today the word "compression" in the golf ball industry relates to a value expressed by a number in the range from 0 to 200 that is given a golf ball. This number defines the deflection that a golf ball undergoes when subjected to a compressive load. Compression simply measures how much the shape a golf ball changes under a constant weight.
Measurement: All three-piece balls and some two-piece balls are measured for compression. A ball that doesn't compress is rated 200; a ball that deflects 2/10ths of an inch or more is rated zero. Between those two extremes, for every 1/1000ths of an inch that the ball compresses, it drops one point from 200 and the compression rating is then established.
Most balls have compression ratings of either 80 [compresses 120/1000th of an inch?], 90 [compresses 110/1000th of an inch?], or 100 [compresses 100/1000th of an inch?]; the lower the compression, the softer the feel. Not every ball marked 80, 90, or 100 is exactly that rating. The actual rating can fall roughly within 3-5 points on either side of the indication. Any ball that falls out of this range is usually sold as range ball, or as X-outs.
In the above Titleist patent, the compression of the family of balls ranges from 60 to 105 Atti, with Cover Hardness (Shore D) measures ranging from 45 to 75. On the Shore A scale, balls are required to have at least a hardness rating of 85 (Shore A durometer) or greater. The Shore A-to-Shore D conversion is not straight-forward, but in general Shore A 85 falls on the Shore D scale between 30 and 60 at the minimum.
The cover design in Callaway's
US Patent 7494428 (Feb. 24, 2009) (
pdf) is described:
"The core preferably has a diameter in the range of 1.610 inches to 1.670 inches. The cover is preferably about 0.015 inch to about 0.045 inch in thickness. Together, the core and the cover combine to form a golf ball preferably having a diameter of 1.680 inches or more, the minimum diameter permitted by the rules of the United States Golf Association and weighing no more than 1.62 ounces.
The cover comprises at least one material selected from the group consisting of polyurethane, polyurea, polyurethane ionomer, epoxy, and unsaturated polyesters, and preferably comprises polyurethane. The material of the cover preferably has a flex modulus in the range of 5,000 to 310,000 pounds per square inch (psi), a Shore D hardness in the range of 20 to 90, good durability, and good scuff resistance and cut resistance. As used herein, polyurethane and/or polyurea is expressed as polyurethane/polyurea.
....
The polyurethane which is selected for use as a golf ball cover preferably has a Shore D hardness (plaque) of from about 10 to about 55 (Shore C of about 15 to about 75), more preferably from about 25 to about 55 (Shore C of about 40 to about 75), and most preferably from about 30 to about 55 (Shore C of about 45 to about 75) for a soft cover layer and from about 20 to about 90, preferably about 30 to about 80, and more preferably about 40 to about 70 for a hard cover layer.
The polyurethane which is to be used for a cover layer preferably has a flex modulus from about 1 to about 310 Kpsi, more preferably from about 3 to about 100 Kpsi, and most preferably from about 3 to about 40 Kpsi for a soft cover layer and 40 to 90 Kpsi for a hard cover layer. Accordingly, covers comprising these materials exhibit similar properties. The polyurethane preferably has good light fastness.
....
The thickness of the cover preferably ranges from 0.015 inch (0.0381 cm) to 0.045 inch (0.1143 cm), more preferably ranges from 0.020 inch (0.0508 cm) to 0.030 inch (0.0762 cm).
The Shore D hardness of the golf ball, as measured on the golf ball, is preferably between 40 Shore D points to 75 Shore D points, and most preferably between 50; Shore D points and 65 Shore D points. The hardness of the golf ball is measured using an Instron Shore D Hardness measurement device wherein the golf ball is placed within a holder and the pin is lowered to the surface to measure the hardness. The average of five measurements is used in calculating the ball hardness. The ball hardness is preferably measured on a land area of the cover. The preferred overall diameter of the golf ball is approximately 1.68 inches, and the preferred mass is approximately 45.5 grams. However, those skilled in the pertinent art will recognize that the diameter of the golf ball may be larger ( e.g. 1.70 inches or 1.72 inches) without departing from the scope and spirit of the present invention. Further, the mass may also vary without departing from the scope and spirit of the present invention. The golf ball preferably has a PGA compression ranging from 50 to 70, and a coefficient of restitution ranging from 0.78 to 0.81."
From
G. Mangum, Dimple Error in Putting:
Dimples and the Putter face -- the Compression of the Dimple at Impact
An additional advantage, at least on short putts, is that if the ball is struck on the seam by the putter, there is no dimple to possibly cause misdirection. As Dave Pelz has shown (see p. 211 in the Putting Bible), when the ball is not compressed, as is the case on very short putts, hitting the edge of a dimple can cause the ball to come off the face at odd angles, odd enough to miss. What Pelz does not say is that the new soft inserts, which allow the face to compress on short putts rather than the ball, takes care of most of this problem. But just to be safe, one could try to strike the ball on a seam or a flat place on the ball's surface between dimples. Pelz advocates marking such a spot on the ball.
Dr Putt, Balance and Seams and Putting
Dave Pelz in his Putting Bible (sec. 9.10, pp 208-211) describes dimpled balls as balls with feet that roll oddly when hit with a putter. The chances of hitting a dimple edge at an odd angle to the intended line of the putt increases with dimple size. The chances that the off-line impact on a dimple edge will actually result in the ball starting off on a line not intended (other than the one that results from the putter face movement at impact towards the center of the ball) increases with the hardness of the cover material in that the cover material hardness decreases the compression of the material from the low-velocity impact of a putter face strike. He shows patterns of dimple impacts on impact tape on the putter face for various putterhead speeds at impact (as indicated by distance of putts on the same green speed), and these patterns indicate that edge-only impact survives incomplete compression of the edge during impact up to a 10-foot putt, and the dimple is completely compressed to flush impact by a 30-foot putt or longer. (Pelz does not specify the cover material, dimple shape or size, point of impact on the dimple, putter face material or green speed he used, so his data is merely suggestive.) Pelz shows a chart he made correlating percent of dimple compression against distance of putt as resulting in inches offline at the end of the putt, again without specifying his materials and conditions. Pelz writes that the dimple-error problem only matters on "short or downhill putts on fast greens". His chart indicates more specifically, however, that the only time the dimple impact causes a miss at the hole is when the putt is long enough (3.52 feet) and the percent of compression low enough (37.5 to 50 percent) or for a longer putt (4.94 feet) with the compression percent being 37.5, 50.0 or 62.5 percent. Pelz seems to imply that putts longer than 4.94 feet do not suffer from sufficiently minimal compression to miss, but this is very unclear. The chart clearly indicates that putts under 3.52 feet do not have a big enough error to miss the hole even at minimal compression, but this assumes dead aim and otherwise perfect impact dynamics.
D. Pelz's Putting Pible fig. 9.10.5, p 211.
Dr Norman Lindsay and Dimple Compression
"It is found that dimple errors are significant for impact deformation depths of about 0.15 millimetres, whereas the errors with impact depths of 0.4 millimetres to 0.5 millimetres or greater are negligible."
Lindsay UK Patent Application GB 2 364 651 A (published 06.02.2002), page 10, pdf sheet 21 of 48.
"The following formula gives a fairly accurate relationship between the maximum depth (millimetres) of a footprint and its span (millimetres) for golf balls:
footprint depth = 0.006 x (footprint span)
Here, the footprint span is taken to be equal to the diameter of a circular footprint that would be obtained with a flat putter on a smooth-durfaced golf ball. Thus, a footprint having a span of 5 millimetres (typical of a short putt with a hard covered golf ball) has a maximum footprint depth of only 0.15 millimetres. It has been found that dimple errors reduce to negligible levels with footprints having spans above 9 millimetres. A span of 9 millimetres equates to a footprint depth of 0.486 millimetres, and from this it can be determined that there is advantage in limiting ridge-depth to between 0.4 millimetres and 0.5 millimetres."
Lindsay UK Patent Application GB 2 364 651 A (published 06.02.2002), page 16, pdf sheet 27 of 48.
Striking a ball to impart an initial velocity of about 3 m/s with a flat-faced putter creates an impact footprint 7 mm in diameter and with a 1.4 mm spaced ridge-face creates a footprint 8.3 mm in diameter, indicating 40% deeper penetration.
Spacing width of ridges at 1.4 millimetres increases depth of penetration 40% to 0.41 millimetres with 6 ridges over that of flat-faced putters, whereas a 1.0-millimetres spacing increases depth of penetration only 19% to 0.34 millimetres with 8 ridges.
Lindsay UK Patent Application GB 2 364 651 A (published 06.02.2002), pages 17-19, pdf sheets 28-30 of 48.
Flat-faced impact at 2.5 m/s on Surlyn balls has a standard deviation in the dimple azimuthal error of 0.66 mm, whereas the ridge-face with 1.4 mm spacing reduces the SD to 0.40 mm. Dunlop DDH 110 ball at 90% confidence level has a SD between 0.69 and 0.75 mm with a flat-faced putter, and the ridge-face putter reduces SD at least 15% and up to 40% normally. Ridges have a Shore D hardness of at least 99 Shore D.
Lindsay UK Patent Application GB 2 364 651 A (published 06.02.2002), pages 25-30, pdf sheets 36-41 of 48.
Dr Norman Lindsay, the Dimple-Error, Ridged Face, and Dwell Time
NESTA's support is enabling Dr Norman Lindsay to solve the problem of dimple-error in putting and pioneer a breakthrough in the spin-imparting properties of golf clubs, improving ball trajectory and length.
NESTA - Norman Lindsay awardee profile -- the dimple problem in putting
All-TS putters are designed to address this problem, featuring a patented face with fine horizontal ridges that distribute the impact force across the dimples of a golf ball to improve line accuracy.
Lindsay Topspin putters have horizontal ridges on face to distribute impact evenly across dimples for better line accuracy
Norman Lindsay and the Theory of Dwell Time
Dwell time' or contact duration between two colliding objects such as a golf club and ball is a well-understood topic in the science of contact mechanics. The founder of contact mechanics was Heinrich Hertz, a brilliant young German physicist. Heinrich Hertz 1857 to 1894 In 1882, Hertz was only 24 years old and working as a research assistant in Berlin University when he published a paper describing his theory of impact. This theory predicts what happens when objects collide and bounce off each other - how much deformation occurs, how the impact force varies with time and the total duration of contact, or what some putter manufacturers call 'dwell time'.
A recent paper in Science and Golf IV by Professor Ieuan Jones of Flinders University presents convincing evidence that when you hit a golf ball, the force and duration of the impact obey the Hertz theory very accurately. Jones studied ball impacts over speeds corresponding to a gentle 'tap in' with a putter and up to a full drive down the fairway. Over this range of speeds the impact duration varied from 0.85 milliseconds for a gentle tap-in to 0.37 milliseconds for a drive.
So the rule is, the faster the swing speed, the shorter the dwell time. The 'dwell time' for a drive is just less than half that for a gentle putt, even though the ball speed off a driver is about 50 times faster than a tap-in. On the putting green, the variation in dwell time for different putt strengths is very much less. For example, on a level green, with the same putter and the same ball, a 10-foot putt will have just less than 15% more dwell time than a 40-foot putt.
The Jones study focussed on how accurately the Hertz theory predicts impact dynamics of one type of golf ball. The ball-hitting implement used in his experiments was made of stainless steel but was very much heavier than a putter head. Replacing this with an average weight putter head would reduce the values he obtained by about 6%. However, different weight putter heads do not change dwell time by much. The dwell time for a 450 grams putter head is only 3% greater than for a head weight of 250 grams.
The property of balls and putters that makes the most difference to dwell time is their hardness. Balata covered balls and elastomer face inserts give longer dwell time than harder materials such as Surlyn or steel, but even these soft materials do not increase dwell time significantly. The Hertz equations use basic elastic constants (Young's modulus and Poisson's ratio) whereas golf balls and putter inserts are usually specified in 'Shore Hardness' scales measured by a hardness tester called a 'durometer'.
This makes it difficult to apply the Hertz equations directly.
What we do know is that with any flat metal-faced putter, the dwell time is almost entirely determined by the ball material so there is no measurable difference between the dwell time from 'soft' metals like aluminium or copper and 'hard' metals like stainless steel. This is because metal putter faces are much harder than golf balls and all the impact deformation occurs in the ball. The Hertz equations also tell us that replacing a metal putter face with an insert made of the same material as the golf ball cover increases the dwell time by only 32%.
Since The Rules of Golf prohibit inserts that are softer than a golf ball, it is very unlikely that 'legal' inserts could increase dwell time by more than 40% to 50% compared to the value obtained with a steel face and a balata covered golf ball. A way of getting round the rules would be to produce a very soft-covered golf ball - much softer than alata - but his would be almost unplayable. To prevent this anomaly, a specification that face inserts must be no less than 85 on a Shore A durometer scale is included in A Guide to the Rules on Clubs and Balls.
MEASURING DWELL TIME
The two traces plotted on the left show the C-Groove accelerometer signals for two different types of ball. Both balls were putted very close to the sweet spot with putt strength of one Stimpmeter¨. In other words, with this putt strength, the balls would roll about 10 feet if the 'green speed' were 10 feet. The top trace is for a Surlyn-covered ball, which gives a short dwell time of about 0.6 milliseconds. The 'ripple' on this trace is caused by putter head vibration. This would disappear if you could get the impact perfectly on the sweet spot, but this is very difficult and the hard covered ball shows up any tiny offset. The lower trace shows the C-Groove deceleration pulse when a balata-covered ball was used. Here the dwell time has increased to about 0.8 milliseconds.
TaylorMade are correct to assume that the 'nubs' on the putter face increase dwell time slightly. It depends on the softness of the insert material and the additional compliance provided by the voids between the nubs. We found that the dwell time for the Nubbins putter was about 0.9 milliseconds, fractionally longer than for a balata ball off a flat steel putter and with the same strength putt (one Stimpmeter¨). However, the claim that this improves ball roll is pure fiction. The nubs in the Nubbins putter are unfortunately similar to dimples on a golf ball and cause line errors. A special groove configuration on Lindsay putters is designed to reduce these line errors, but the TaylorMade nubs will tend to slightly increase dimple error effect. (More details of this error effect will appear in future editions of this website.) As it happens, the Nubbins putter is probably less prone to dimple error effects than some other putters because its soft insert material helps to reduce dimple errors.
So what is the maximum dwell time you can get from a 'legal' putter? This is going to be from a putter with an exceptionally soft insert - probably softer than a standard balata-covered ball. One likely candidate is a Fisher putter. Fisher claim that their putters have more than twice the dwell time of a flat-faced steel putter and quote some quite amazing 'scientifically proved' figures. Undoubtedly, their putters have very soft inserts Ð in fact just within the limits set by The Rules of Golf Ð but their figures for dwell time are highly exaggerated and again, equating dwell time with overspin is simply marketing hype.
Top trace: The Nubbins putter - dwell time just less than one millisecond.
Lower trace: Fisher F-6 putter.
Dwell time marginally over one millisecond.
(Both putters tested against a balata-covered ball at one Stimpmeter¨ putt strength.)
Fisher Golf's claimed dwell time
Measuring the hardness of the Fisher putter with a durometer. Its hardness measures about 87 Shore A.This is within the legal limits set by The Rules of Golf but quite a bit softer than most 'soft-covered' golf balls.
Zwick Durometer -- Shore D
Comparing Shore A and Shore D is not straight-forward.
Norman Lindsay -- Dwell Time in Putting
Dr Putt and Ball Balance
Dr. Putt has tested a variety of new and used balls to see how frequently they are in or out of balance. Here is what he found. ALL the used balls in his bag were out of balance. So once played with for a few holes, the ball is likely to become out of balance.
On new balls, the results were mixed. The relatively inexpensive Top Flight XL-3000's were out of balance more than half of the time (9 of 15 were out). Dr. Putt also tested a sleeve of the very expensive Titleist Pro V1's and all three were out of balance. Then he tested the Wilson Staff True Distance ball, which is advertised as being in balance. And all three in the sleeve tested were in balance. Will they stay in balance? Dr. Putt will report on this later, retesting the balls after each 9 holes played.
What is the bottom line on ball balance? Most balls you buy or play with are probably out of balance. And as Pelz has shown, it can make a difference. The difference is most obvious on very fast and smooth greens where other factors like grain and bumps do not mask the effects of balance. So if you are in an important round of golf on fast and smooth greens, play with balls that are in balance or have their heavy point on the seam. Then you can draw an aim line on the seam and feel much more confident about getting a true roll."
Once a ball has been played a significant number of full hits, the ball is probably deformed in shape and also in mass distribution. Pros toss away new balls after a mere three holes.
Ball balancing as is done by the Check-Go ball balancer does not really address the issue of "how much" out of balance a ball might be. Spinning a ball only accomplishes locating the out-of-center Center of Mass in a single plane, marked by a circle around the plane with a felt pen. This pen circle does not indicate either which direction the COM extends away from the center of the sphere OR how far out of the center the COM is located. The latter determines "how much" the ball is out of balance. To test "how much" a ball is out of balance requires observing the "pendular" timing of the swing of the COM beneath the sphere center as the wobbling / floating ball settles down after spinning. Once the ball is no longer spinning end-over-end but is swinging to and fro like a pendulum, the timing of the swing from one side to another indicates the extent the COM is out of the sphere center. The MAXIMUM out-of-balance is certainly less than the radius of the ball, 0.84", since the deformed COM cannot reach the exterior of the sphere without completely displacing one half of the ball to the opposite side. A realistic maximum is probably closer to half the radius, 0.42". A "pendulum" of length 0.42" has a timing on earth of 0.104 seconds. The maximum is the slowest swinging from one side to another, and any ball whose "wobble" settles down to the same top spot repeatedly with a swing time of 1/10th of a second is seriously out of balance. A COM half that far out of the center (0.21") has a timing of 0.074 seconds, which is nearly the same time it takes for a ball to drop one-half of its diameter. Even this out of balance is substantial.
CONCLUSION
The delofted putter strikes the ball very near the equator and for most putts, putters and balls, the putter face effectively crushes the dimple edge out of significance. Some ridges help, as does a soft insert, but generally the dimple is not a big source of error. An out-of balance ball is a significant source of error, as this source contributes cumulatively, and is therefore worse on longer putts and on faster green surfaces.
Concerning the launching of the ball off the ground, the physics is the vertical component of the impact vector in relation to the mass and weight of the ball, but this is somewhat complicated. The impact vector STARTS aiming thru the spherical center of the ball and then either remains aimed that direction throughout impact duration ("dwell time") or changes the vector direction. Typically, in a pendular stroke, the vector starts thru the ball's center and then shifts to a vector aimed above the ball's center, as the putter face gains loft and swings upward on the pendular arc. The "resultant" vector then that determines the lifting force is aimed somewhat above the center of the ball in such a case. Since the dwell time is brief, on the order of 0.005 seconds, the putter face's trajectory does not add much loft or travel far along the upward arc thru impact.
For example, a putter face moving along the line of the putt at 100 inches per second (about 250 cm/s), which is fairly typical for putts between 60 inches (5 feet) and 120 inches (10 feet). Such velocity for 0.005 seconds traverses a mere 0.5 inches. That is, for putts in this range, the ball and putter make contact at then separate after 1/2 an inch. A putter face that swings on a 54-inch pendulum that adds loft over 0.5 inches will add 0.53 degrees and rise above the ground 0.002 inches. Assuming the "resultant" force has an angle HALF this change / addition, then the resultant angle is the dynamic loft at impact onset plus 0.53 degrees for the addition during impact. Assuming a 3-degree putter with the back of the ball 1 inch ahead of the bottom of the stroke with "dwell time" of 0.005 seconds, the sole rises 0.005 inches and the 3-degree putter gains 0.8 degrees loft, making impact at 3.53 degrees 0.0015 inches higher than address and finishing contact with the ball at 3.8 degrees at the sole 0.0052 inches higher than at address. The "resultant" is somewhere around 3.66 degrees, aiming from a spot below the equator that started at a point on the face 0.0015 inches lower than at address on the back of the ball.
Whether a putt force at this angle thru the ball is sufficient to "launch" the ball off the ground significantly so that the ball begins to bounce along the path depends upon the putter head mass and velocity thru impact. Whether the ball "launches" off the ground or whether the impact "knocks a top rolling over a bottom" of the ball is a matter for detailed experimental exploration. My sense of experience is that putts in the range of 2-5 meters with typical putter head masses and green surface speeds do not result in the ball "launching" significantly when the dynamic loft at impact is under 5 degrees, at least when the COM of the putter head is not substantially above the impact point of ball on the face. For putts longer than this, most typical putter designs will launch the ball 3-7 inches off the ground at the start, and sometimes this beginning generates a persistent bouncing along the path. I believe this depends to a large extent upon the "quality" of the surface, in terms of its "trueness" and uniformity and smoothness.
Ultimately, though, balls DO get launched off the ground, so the idea that "all balls effectively are impacted on the equator" doesn't solve the issue. Nor does this idea remove the issue of whether hitting the ball above or below the equator, however small the distance above or below, does not significantly cause or influence the launching of the ball. There is a lot more work to do on this.
Cheers!
Geoff Mangum
Putting Coach and Theorist
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