Blade Performance Vs Wood Type and Design

Blade:             < 80 grams light,  80- 90 grams medium,   91+ grams heavy

Racket:            < 175 grams light,  175 – 188 grams medium,  189+ grams heavy

Many factors effect blade performance: mass, bending frequency, damping, swing weight, and surface layer hardness.  All of these factors also have to be matched to your personal preferences and style of play.  Of these factors, only mass is ever directly listed in catalogs.  Surface layer hardness is alluded to as soft, medium or hard feel.  Bending frequency and damping are mixed together and alluded to as defensive (low frequency, higher damping), all-round (medium frequency, unknown damping), offensive (high frequency, low damping). 

When I started thinking about building a custom blade, I started with a table of wood engineering properties on about 200 domestic and imported species commercially available that the U.S.   I then calculated a simple figure of merit composed of the Modulus of Elasticity (in Mpa) divided by the Specific Gravity (a measure of density) to highlight woods that were elastic and light in the hope that this would identify species that could be good for a table tennis blade.   My results included many woods that are found in table tennis blades so my approach seemed pretty good!  A larger database would probably turn up additional candidate woods.

Now if we want to figure out how stiff we want to make the blade we either have to know how to calculate what stiffness we want (hard) or we just have to make one and see how it feels.  If it plays too stiff, then make a second one thinner, too flexible, then make the next one thicker.  Once you learn what thickness you like in a blade with a given wood, (thickness is the biggest factor in stiffness), you’ll probably like the same thickness in most every wood with similar engineering properties.  Since none of the listed soft woods are dramatically different from each other, the solutions to what a good feeling single ply blade out of any of them will end up being pretty close to each other.

Now as table tennis players, we already know that most single ply blades are 8 - 10 mm thick Hinoki.  Some are made from Port-Oxford Cedar, or other cedars or spruces or other soft woods but they all seem to end up being 8 – 10 mm thick.  This is because each of the these softwood species exhibit similar engineering properties.   This is why you don’t see anyone posting how great their single ply 5 mm fir single ply is and how it hits faster than a Schlager Carbon!

But what if I want a hard stiff blade for hitting but I don’t’ want a blade 10 mm thick?  We also know that most table tennis blades are 5 ply or 7 ply and that many have synthetic layers (carbon, carbon-aramid, etc) or are even more complex.  Why?  The answer to these is based on beam theory and how material selection and distribution through a blade establishes the bending properties (defined as the second moment of inertia) of the blade.   These more complex construction techniques let us make the blade stiffer and/or more uniform in its bending properties while keeping its mass or thickness the same or even less than could ever be achieved with a single ply of wood.

I’ll address this in a subsequent post…………

In my earlier post I said nobody posts about their 5 mm single ply blade and one reason they don’t is because such blades are easy to break in play.   Wood’s physical properties are non-isotropic, i.e. they are not the same in every direction.  If you look at pictures of a single ply blade, you always see parallel grain lines running the length of the blade from the handle to the tip.  This aligns the strongest axis of the wood with the bending that occurs during table tennis.  If instead you were to grab the single ply blade by its two sides edges and bend it, the wood is much weaker when bent on this axis and it can easily split along one of the grain lines.  You might even be able to snap it with your hands.

So lets we want to use a particular wood but we also want to make its bending properties from side to side the same as from handle to tip.  This is done by making plywood from two or more thinner layers of the wood glued together with the grain directions placed at an angle with respect to each other.   If we have three layers, then we glue them at 60 degrees to get the most uniform bending in every axis (and so on).  We could use smaller angles to retain greater bending strength in a preferred direction while still improving strength in the perpendicular direction.  Our plywood will have more uniform bending properties but it won’t be as stiff as the original wood was in its strongest direction since we’ve distributed the total stiffness in other directions as well. 

The glue type, solids content, and pressure also matter when making plywood as the glue joint also has to flex when the plywood flexes.  Hide glue (made from boiled animal tendons) is the traditional wood glue used for centuries in furniture, musical instruments and – custom table tennis blades!   Few blades today however are made with hide glue.  Hide glues are flexible and a hide glue joint can be fully undone and repaired with steam heat.  Modern glues such as PVA (white or yellow wood glues) are rubbery synthetic polymers that are good for porous materials such as wood and paper and they are commonly used in making books because of their flexibility.  PVA glue and epoxies dominate table tennis blade manufacturing today.  These also can be very good glues and are very durable. 

Now lets assume we want to make a 4 mm thick three layer plywood out of a harder wood such as black walnut, with a blade face and handle total area 230 cm^2 (Butterfly Amultart size).  Since black walnut has a specific gravity of 0.55 our blade’s expected total weight (less handle pieces) is 0.55 grams/cm^3 x  230 cm^2 x 0.4 cm =  50.6 grams.  Now we add 24 grams for a walnut wood handle, plus 1.3 gram of glue per ply and we arrive at a total weight (in theory) of 77 grams. 

Now, when making plywood, one needs a vacuum press and equipment to evenly sand down the layers to the needed thickness.  This gets more expensive than just buying and trying blades but it lets you build blades you can’t buy, but this cuts both ways since you can also buy blades you can’t build.   

If we play with our pretty thin blade, it should feel hard on ball contact because black walnut has a side hardness of 4500 N (compared to Port-Oxford Cedar which has a side hardness of 2800 N), but since it is only 4mm thick we also might find that the blade itself is more flexible than we expect.  This will be further exacerbated since the wood’s maximum stiffness has been reduced by the averaging that occurred when making it into plywood!     So, if we had picked black walnut in order to make a hitters blade, and assuming we are finding it too flexible for hitting, the best path to making it stiffer is to make it thicker.  

A key mechanical engineering property of uniform materials (such as our plywood) is that stiffness increases with the third power of the thickness.  So if we increase our thickness from 4mm to 5mm, (i.e. a 25% increase in thickness) we end up with a  5³/4³ = 125/64 = 195% increase in stiffness.   Unfortunately, as you can see that with adding only 1 mm more thickness, we’ve gone from a medium 77 grams to a pretty heavy 96 grams and we don’t even know if were stiff enough yet (of course we could very well already be too stiff, but we’re the main point is that working with hard wood only gives you a very short thickness range to work with before you get too heavy.   This is why you don’t see many table tennis blades made from hardwoods – hey do exist but they are rare.   As an example, look how wide the various usable thicknesses are for Sitka Spruce are and how much the stiffness varies over that thickness range compared to black walnut.  The walnut gets too heavy before its stiffness has even doubled.

Remember stiffness increases as the third power of thickness…

I haven’t gotten to beam theory yet other than the point on stiffness being primarily driven by beam thickness, but I’m getting there.    I’ll have another post later……

Blade Performance Vs Wood Type and Other Design Issues: 

Solid mechanics uses non-linear matrix models to describe physical behavior but this math is often simplified into linear models, to ease computation.  Beam Theory is one such simplification. However, non-linear models are becoming more common and easier to work with due to computers.  Most real world physical phenomena start off linear during small changes but become non-linear as changes grow.  Common types of behavior under deformation are:

1. http://en.wikipedia.org/wiki/Elasticity_%28physics%29 - - Elastic   – When an applied stress is removed, the material returns to its undeformed state. Linearly elastic materials, those that deform proportionally to the applied load, can be described by the http://en.wikipedia.org/wiki/Linear_elasticity - equations such as http://en.wikipedia.org/wiki/Hookes_law - , and the Euler-Bernoulli Beam Equation.
2. http://en.wikipedia.org/wiki/Viscoelasticity - - Viscoelastic – These are materials that behave elastically, but also have http://en.wikipedia.org/wiki/Friction - : when the stress is applied and removed, work has to be done against the damping effects and is converted in heat within the material resulting in a http://en.wikipedia.org/wiki/Hysteresis_loop - in the stress–strain curve. This implies that the material response has time-dependence.
3. http://en.wikipedia.org/wiki/Plasticity_%28physics%29 - - Plastic – Materials that behave elastically generally do so when the applied stress is less than a yield value. When the stress is greater than the yield stress, the material behaves plastically and does not return to its previous state. That is, deformation that occurs after yield is permanent.

Computer Modeling Tools:  The best way to explore this topic is to model the table tennis blades within a Finite Element Analysis program capable of dealing with composite material construction, dynamic analysis and impact deformations.  With such software one could model various table tennis blades, calculate their different frequency responses and bending modes and even cover them with inverted rubber and sponge hardness to study sponge hardness versus blade hardness, how striking method and blade frequency effects power and spin creation and so on.   Unfortunately, I do not have such software and though I am trying to get some, actually getting it, learning to use it and finally building an accurate physical model will take months of effort.  If I am successful on this longer term path, I will be sure to come back.

 

Still, no one can build a good model without knowing the theory first.  So lets start with the Euler-Bernoulli Beam Equation (first written around 1750) for the elastic bending of a 1-dimensional beam.  This 4^th order differential equation describes the relationship between a beam's deflection u and the applied load w.

 

 

In this equation, the deflection at each point x, is the function u = u(x,t).   The load at point x, is w = w(x,u,t,….) where w can be function of x, u, time, or other variables.  E is the http://en.wikipedia.org/wiki/Elastic_modulus - of the beam material and this is the same as the “modulus of elasticity" that we saw with wood species earlier.   I, the http://en.wikipedia.org/wiki/Second_moment_of_area - , and this is a value that accounts for how the beam’s cross section shape resists bending.  We’ll come back to E and I later.

To simplify our math (by a lot), we will mentally reshape our table tennis blade into 1”x7” inch strip with one end clamped into a vice so that a 1”x6” inch blade face strip sticks out and is free to vibrate.  This gives us a classic cantilever beam built up as a single ply, a 5 ply, a carbon blade of thickness ”h”. 

Our simplifications make u = u(x), http://en.wikipedia.org/wiki/Stiffness - - E - I constant, and our beam mass density r, constant.  We now pick the load w(t) so that we start with the beam equation for a vibrating cantilever:

 

The following graphs show our cantilevered blade vibrating in a single mode.    Interestingly, at the mid-point of every vibration mode, the blade returns to flat.

 

 

 

When we play, our hands feel the blade vibrations caused by the ball impact, (i.e. we feel the vibrations of our cantilever).  When we hit the ball on the sweet spot, we mainly feel the dwell and catapult from the mode n = 1 vibration, the fundamental frequency of the blade.  When the blade is soft or stiff, we feel that softness or stiffness as more or less dwell.  More dwell means a lower mode 1 frequency while less dwell means a higher mode 1 frequency and a stiffer blade.

Besides the mode 1 behavior we normally feel when playing, we sometimes hit the ball off center.  When we hit the ball hard and off center, we will also feel some higher frequency vibrations due to higher frequency vibration modes being excited by the higher energy impact. 

Most of us have had the experience of hitting a ball hard but missing the sweet spot of a racket or bat or club.  When this happens, the impact drives a lot of energy into the higher frequency modes of the bat or club.  That’s why your hands sting when you miss hit a baseball or a golf ball.  High frequency modes start vibrating and your hands damp it out.

The last thing we need to do now is learn at how to calculate E and I for layered beams.  We didn’t have to worry about how to do this for our single ply blades but we have to worry about it for multi-ply blades.  When using layered beams in which the plies are oriented symmetrically about the midplane and where the orthotropic axes of material symmetry in each ply are (ex. the wood grain is) parallel to the beam edges, can be analyzed by via classical beam theory if the bending stiffness EI is replaced by the Effective Stiffness of the beam.

 

 

Now the moment of inertia I for a beam with rectangular cross section of width b and height h, is given by I = b h³/12.   

 

When this cross section is not at the center of the beam (such as will be the case with all but the center ply in our laminate table tennis blade), the Parallel Axis Theorem lets us calculate the moment of Inertia of each ply.  Thus the moment of inertia I ^k for the k^th ply is then:

 

I ^k = b h³/12 +  bh d ² 

 

where d is the distance from the midplane to the center of the k^th ply.

 

We can now update our frequency versus stiffness relationship to account for the effective stiffness of a laminate blade so that we are finally ready to calculate how stiffness and frequency behavior is affected by material selection, thickness and construction techniques.  Our frequency vs stiffness equation for multilayer construction blades is:

 

 

Ok.   There is a good bit we can learn about blade cross-section design by studying this formula and even more by calculating how different blade designs behave.  Even with all this work though, we’ve only studied a strip of wood constructed like a blade.  A real blade has a larger oval face and a throat that also greatly affects its stiffness and frequency behavior.  Getting a handle on this requires numerical modeling.

 

This has been a long hard post, so I’ll save the looking and calculating for the next post…..  I’ll be back with Part IV……..

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Galaxy T1 89 gm

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Donic Acuda S2, Max, Red

Blade Performance Vs Wood Type and Other Design Issues:

In the table below is some of the engineering data that I showed in the Part I post, and I’ve also added some engineering data for some more woods and some carbon fibers.   All of the calculations pull from the data below.  I don’t use all of it but I do calculate 11 different types of blades.

Now it has been pointed out that you can’t always calculate everything and that’s true.  More importantly, not every piece of wood is the same even if it is from the same species and area.  For example, I don’t think I was able to find good data for Hinoki, and it certainly does not appear to represent the properties of Kiso Hinoki as the Port-Oxford Cedar seemed to out perform the Hinoki fairly easily.  Hinoki has a pretty wide range of density so this is to be expected of it (and this is to be expected of all woods).  

Similarly, all carbon fibers are not the same, and which types of carbon fibers you pick and if you use them as a woven cloth or as layers of uniaxial fibers at different angles strongly effects your results. 

You’ll see in my last blade calculation where I calculate for a Schlager ULC Carbon.  This is a Schlager Carbon construction where I have instead aligned all of the uniaxial carbon layers with the axis of the cantilever.  This orientation really increases the stiffness and frequency while reducing the normally larger sweet spot size of a carbon blade to that of a wood blade.  This table shows the frequency (speed) results for a lot of different blades.

You can see from the Hinoki single ply results that stiffness does increase as the 3^rd power of thickness.    Ex.  10³ / 8³ x 9.764 = 19.0625, i.e. very close to 19.071.  HOWEVER, stiffness does not equal SPEED, it contributes to speed.  

FREQUENCY however is directly tied to SPEED but none of the blade manufacturers measure the frequency of their blades.   You can listen to the frequency and pick the faster blade by listening for the higher pitch. 

Here are the rest of the blade calculations I’ve done: 

OK that should wrap up Part IV…. The calculations do show that changing design does change frequency behavior and therefore blade speed.  There is of course other things to worry about like the size of the sweet spot and touch at low speeds but that as I said in the beginning, all of this is very complicated and just getting even a basic handle on the simple parameter of blade speed (frequency) has taken some work.  

In any case, while there is more to consider perhaps this is enough….

Part V  -- TO COME  -

A poster cautions:  "Be careful when applying purely mathematical considerations to wooden materials. Pythagoras tried to do that to stringed instruments. However, every stringed instrument using Pythagorean tuning has a few dead or flat notes, even holding "perfect" tune. Feel is much the same way, defying quantification."   
 
Answer:  Wooden materials are not better understood by not studying them.   Furthermore the entire subject of the thread was the question of what are the playing characteristics of blade materials.  
 It is far better to have limited knowledge than to decide to have none because you can't have it all. Thus it follows that a Pythagorean tuned instrument is better than an instrument that has no tuning at all.
 
Now to your point, this exercise of understanding how a blade's wood selection and design construction effects its behavior doesn't relieve one of the need to build a blade expertly nor does it relieve one of becoming an expert player to expertly match a blade to your game. 
 
Furthermore, unit to unit differences in wood and production variations always result in every blade and every rubber sheet being different as well.   Many expert players weigh their blades (or stick with the same blade for years) and weighing their rubber sheets as a way of controlling the repeatability of their equipment.  
 
We have all seen 1000's of posts on the topic of equipment selection... "Is this a good blade/rubber...." and what's good for this or that and all of the opinions and stuff.  And we all look at and wonder what the professionals are using and are they really using that and why are they using that and should we be using that and if we were using that would it make us better and can we even get the same stuff the professionals are using and on and on and on and on and on.......
 
So right now I am pretty happy to have a reasonably well founded equation and even if its not any better than Pytharoras' tuning, if it tunes most of the notes pretty well, then I will have learned something concrete, and thats better than the current state where we spend most of our time arguing about what also floats on water besides witches....

AgentHEX wrote: I believe the spring of just the ball itself is adequate to explain this. After all it bounces quite well on a infinitely thick/stiff surface......

The mode n=1 equation is:

JRSDallas wrote: I use blade frequency as an predictive measure for blade speed since (1) it is tied to system stiffness, i.e. to the (square root of the) ratio of elastic modulus divided by the density

1              If blade frequency can be used to predict blade speed because of its relationship to system stiffness, then does it follow that calculating stiffness will also give us a measure of blade speed?

2              The formula quoted above uses “density”.  Are you using average dried weight (measured in kg/m³) rather than specific gravity?

3              How is the formula applied?  (I said I had trouble with numbers!)  For example, if I know that PO cedar has an average dried weight of 430 kg/m³ and MoE of 11.35 GPa, then stiffness = √(430/11.35) ?

Again, I’m sorry if I have no idea what I’m talking about, but I would be really interested to know if this produces some meaningful way of comparing different timbers.  Thanks.

2              The formula quoted above uses “density”.  Are you using average dried weight (measured in kg/m³) rather than specific gravity?  I use any weight reference I can find (dried, specific gravity, etc) and convert to the units needed.  I don't use green weight as we don't use green wood in our blades. 

3              How is the formula applied?  (I said I had trouble with numbers!)  For example, if I know that PO cedar has an average dried weight of 430 kg/m³ and MoE of 11.35 GPa, then stiffness = √(430/11.35) ? 

You can see the spreadsheet tables of calculations that I did for various rectangular approximations to blades earlier in the thread.  However, I have only done these calculations for that original thread in late 2008.  I did it at the time to illustrate that the formula results made sense, i.e. A 10mm single ply hinoki is fast - Yep the calculations predict that.  An 8mm single ply hinoki is pretty slow - Yep the calculations predict that.   The Schlager Carbon is pretty fast - Yep the calculations predict that.   If you are designing a blade that you will build but want to optimize something, then go ahead and work out the calculations for some ply combinations you are considering.  If you understand the equation though, you can see how the ply combinations effect the result and pick what you want to do.  If your goal is to build a blade that is faster than Schlager Carbon, build a stiffer plywood.  The lightest way to do that is to increase the thickness of the light center ply so that the carbon layers are moved further apart.  Or, if you have a fabric with higher modulus than carbon, then use it instead of carbon and don't increase center ply thickness, or do both and get something much faster.  If you don't like how carbon feels but still want a result that is still faster than Schlager Carbon then use Carbon-Aralyte and increase the center ply thickness even more to offset the lower elastic modulus of Carbon-Aralyte.  The formula will help you determine exact thickness changes needed but you already know what to do. 

 Now one can make accurate measures of blade frequency spectrum (handle in clamp, ball bounce at center of percussion, microphone, Fourier decomposition into frequency spectrum, identify spectral peaks - see pictures from the original thread).  I also did that for 30 or so blades in late 2008/early 2009.  I was not looking for the calculated frequency of my rectangle approximation to predict the measured frequency of a real blade with the same plywood, I was trying to see the behavior differences between blades matched what the calculations predicted.  Yes, Schlager Carbon is far faster than Clipper.  Yes, EPOX TOPSPEED isn't very fast even though the manufacturer (and CONTRA) rate it OFF+.    Perhaps I got a bad one?  No.  The review is wrong.  Can you hit a fast ball with Clipper and EPOX TOPSPEED, absolutely but you'll spend more energy doing it.

So my recommendation is to understand the system behavior, not to calculate a particular output that results from particular inputs.

I only bounce a blade (of normal size) on my head so that I can hear its tone (pitch, frequency).  If the tone is higher than another blade, then it is faster than that other blade.  If I am buying out of a catalog, then I look at its overall thickness and the layers in the plywood.   Innerforce ZLC is absolutely slower than Amultart primarily because its ZLC layers are closer to the center and it is thus less stiff. 

AgentHEX wrote: >JRSDallas said:   "If you have two stiff plys, placing them at the outside of the blade increases blade stiffness the most."    (1) This is slightly true, but to a much less extent than most people assume. To understand this intuitively, it's worth pondering what it means for a substance to be "solid". (2) As alluded to before, solids have very strong molecular coupling which make the "stiffness" of the outside very much connected to the stiffness of the inside. That's why plate theory with a simple x^3 relationship works out, with no need to integrate by layer.   (3) When it comes down to collisions it's stiffness that matters: it relates directly to elasticity which is the namesake difference between elastic and inelastic collisions………

Note: I have added numbers and underlining above - JRSD

(1) No it is exactly true and it matters when a blade uses different material layers in its construction.  I calculated an example comparing two blades of identical thickness just two JRSDallas posts ago.   

(2) HEX talks plate theory but does not recognize that the plate theory equation he has seen assumes an isotropic and homogeneous material (i.e. a uniformly thick single material with uniform physical properties in all directions) in order to end up with a simple X^3 relationship.    (Note:  I also assume HEX means x=blade thickness.  In this thread I use h as the symbol for blade thickness).   NOTE: A single ply blade is the closest TT example to a homogeneous plate, but wood is still non-isotropic.  In my original derivation I discussed but then ignored the non-isotropic aspects of wood in order to keep the math simple.  I did however solve for the non-homogeneous lamina case where the blade is made of different layers.

SO………when a plate is non-homogenous lamina (i.e. if it has layers of different materials), you have to account for each layer’s contribution to plate modulus E_blade (and to be actually correct also have to account for each layers moment of area I_blade) so as to obtain an overall blade modulus E_blade for use in a Mindlin–Reissner plate theory formula such as:    

   where:

D = bending stiffness of a plate (also known as flexural rigidity) = E_blade * I_blade.  In the plate theory formula shown above the moment term I, is already captured in the value 2/(3*(1-v^2) so you don't see it in the formula.  Still since the equation is based on assuming a calculation at the center of the plate, it will miss the weighted effect on increases or decreases in E from layers that are not at the center of the plate. 

h = blade thickness

E = Young's modulus (of the plate) = E_blade obtained through summation through the blade layers – (exactly the issue that HEX ignores)

v = Poisson's ratio.  

So in estimating blade stiffness, can we only look at blade thickness h as HEX says?  No, you have to look at both E and h and the plate theory equation clearly shows this.  Different materials have different E, and if one compares two materials of thickness h, the material with the higher E will be stiffer.    Similarly if one compares two blades of equal overall modulus E_blade, the thicker blade will be stiffer.   Finally, for any given blade construction, increasing thickness h very quickly makes it stiffer – i.e. stiffness grows as the third power of thickness (as originally posted 05/22/2013 at 7:24am, paragraph 6).     Now if you are comparing blades made from woods with similar E, that are constructed similarly, they will have similar modulus E_blade, and in such cases, h is the decider as to which blade is stiffer.        

(3)  Here HEX makes the connection that stiffness matters in collisions (finally).  Unfortunately, while h is often known, no TT catalog lists, or TT player measures blade modulus to let us even use the plate theory formula as a practical tool.   SO………how then can a TT player judge blade speed before buying and trying?   

Hmmmmmmm………………….Wait I remember!  Someone in this thread showed using math from the theory of beams (which is also the source of plate theory), that the stiffness of the blade (E_blade * I_blade)  can be judged by the frequency of vibration that the blade makes when struck!   As I remember, that formula also included structural terms that let us understand the connection between the mode frequencies and how the layers of the blade contributed.   Further, I recall that from all of the examples calculated and empirical data presented that it seems to be true that to compare the stiffness of different (typical) TT blades, all we have to do is listen to which blade has the higher or lower pitched sound to know which blade is stiffer (generally observed to be faster) or less stiff (generally observed to be slower).       

AgentHEX wrote: > You're assuming that those frequency calcs are directly correlated which is very much in question (they certainly aren't with respect to size, and who knows what else), ie. a case of begging the question.

It seems that you like to argue (and a good argument can be fun) but your apparent strategy is to mix misuse of concepts with the innuendo and incomplete statements (such as your first sentence above). You also like to put words in the mouths of others even though the facts of what they have presented are on the page.  No matter, I’ll play along.     

1.  You accuse that - "I am assuming frequency is directly correlated which is very much in question".   Well since you have by omission left me with my choice of what frequency is correlated with, I hereby claim that frequency is directly correlated with frequency, i.e. f = f.          I win.

2.  Your parenthetic statement (they certainly aren’t with respect to size, and who knows what else) says you claim that (frequency) is not correlated with blade size or other physical parameters.    I disagree and claim that for a symmetrically laminate cantilevered beam (as a 1-D approximation to a table tennis blade face) exhibiting vibration mode frequencies f_{n ,} THAT frequency is in fact correlated with the causal function:

Blade frequency of vibration mode n vs blade stiffness:    where:

E ^k      is the modulus of the k^th lamina relative to the midplane

I ^k     is the moment of inertia of the k^th laminate layer relative to the midplane, where  I ^k = b h_k³/12 + bh_k d_k ²

d_k     is the distance from the midplane of the laminate beam to the center of the kth laminate ply where k = 1 to N.

h_k    is the thickness of the kth laminate layer

d_k     is the distance from the midplane to the center of the kth ply

b      is the width of every laminate layer

L      is the length of the every laminate layer

N     is the number of layers in the laminate beam

n    is an integer identifying specific modes of vibration where  n = 1, 2, 3…..

r     is the mean density of the laminate beam

A     is the area of the end face of the laminate beam, wher A = width b * sum of h_k, for k= 1 to N

Further the above correlation is causally based on the physical parameters of blade size (parameters length L, width b, cross section face area A), and other physical parameters (density r, Young’s modulus E, Moment of Area I ).    Since this relationship was derived from the (1) accepted theory of beams, and (2) accepted rules for mathematical manipulation, it is widely accepted as correct as it stands.      Finally, as this is the same claim I made in 2008 when I originally posted this derivation and in 2013 when I reposted it, the evidence suggests that your understanding correlates with unskilled.   I win again.

You of course are free to contest this by showing that the theory of beams is wrong (to first order, and not at some limit of relativistic speeds or other silly flapping about), or show that the mathematical solution to the differential equation is in substantially in error.   I won’t hold my breath.

Please also refrain from using Wolfram Alpha to pull up some non-physical function (as you did earlier in this thread) to argue that it is a better predictor.  We all know that the declining number of pirates is strongly negatively correlated with the increase in global temperatures.  Of course the pirate correlation IS true, but only the unskilled believe it to be causal.

3.  HEX said  “To elaborate further, if what you're claiming about the relative significant of surface vs inner plies is true it should certainly show up in this category of theories as some relative weighted factor with respect to distance from center (then integrated over the thickness of the plate).” 

Well lets look….….yes the contribution of plies that are displaced from the plywood mid-plane correctly does show up as a distance squared weighting factor per the Parallel Axis theorem, and by golly the weighing has been there since the original posting of this thread in 2008, and since this MyTT.net reposting in 2013.  Maybe you should get your eyes checked?

So…..just to be clear…..your claim that I was not correct about the impact of ply placement is and has been pointless from the beginning.        I win this one as well – Just saying what’s true.        

To assist you, the detail of how the area moment of inertia for the kth lamination of the laminate beam is calculated below.  The correct weighting of course is there as it was in the beginning and ever will be. 

Finally, I’ve just tonight discovered an article in Procedia Engineering, Volume 34, 2012, Pages 604–609, ENGINEERING OF SPORT CONFERENCE 2012  that addresses the same topic that my original 2008 thread addressed.     The abstract clearly includes the approach I took in my original 2008 work.   Makes me feel pretty damn good that I took this same path six years ahead of them.  --  I think this counts as another win for me.  Just sayin’.  

Vibro-acoustic of table tennis rackets at ball impact: influence of the blade plywood composition

Lionel Manina, Florian Gaberta, Marc Poggib, Nicolas Havardc

Abstract - The performances of a table tennis racket can be qualified with several adjectives like: fast, slow, stiff, adhesive, controllable, etc. These qualifications are subjective since they are relative to the sensory analysis made by each player. It appears that the noise produced at the ball impact on a racket has a great influence on the opinion that a player can give about a racket. Moreover, the sound emitted at the stroke can be appreciated differently among several players. Hence a good sound may give a positive a priori to the player racket appreciation. The work presented first demonstrates the correlation between the acoustic frequency spectrum and the vibration frequency spectrum of a racket following the ball impact. The analysis is first performed on the racket blades without rubbers glued on. The vibration modes that produce the sound at the ball impact were identified experimentally. It is shown that there are two essential modes responsible of the sound emitted. In second, comparisons were made between several rackets composed of different composite plywoods. Their influence on the sound emitted is shown experimentally. Indeed, depending on the wood essences of the different plies, their thickness and their fibers relative orientation the sound produced will be different. Then the influence of the rubbers glued on both blade sides is studied. The vibration modes are the same but the frequencies are lower. The sound can be qualified as sharp, long, clear, deep, hollow, and plain. Some correlations with the player appreciations are made.

References

        [1]      Y. Kawazoe, D. Suzuki  - Prediction of Table Tennis Racket Restitution Performance Based on the Impact Analysis, Theoretical and Applied Mechanics Japan, 52 (2003), pp. 163–174

        [2]     K. Tienfenbacher, A. Durey -  The impact of the table tennis ball on the racket, Int.  Journal of Table Tennis sciences, 2 (1994), p. 1994

4.  Earlier in this thread, you said that the dwell time of a TT ball bounce on a racket was 1ms and that a supersonic TT ball impact on a hard bat and a molecular (hard sphere) model of a TT ball bounce proved this.     You are adamant about 1ms and have used it to argue against popular belief that blade flex and recovery contributes to catapulting the ball off the racket.   I said I thought that dwell was 3 – 5 msec,  but as I could not find my original source that perhaps my memory of an article I saw in 2008 was wrong. 

I have now found the article I saw in 2008.  What is more, it shows table tennis ball on racket contact times (dwell) of 4-5 ms (see Fig 15 below), a marginal 400% to 500% longer than the forcefully claimed 1ms.   The article is in: Science and racket sports III : the proceedings of the Eighth International Table Tennis Federation Sports Science Congress and the Third World Congress of Science and Racket Sports / Published London ; Routledge, 2004.   Y. Kawazoe and D. Suzuki -- Characterization of table tennis racket and sandwich rubbers (Tamasu Corporation aka Butterfly ).  Selected excerpt below – Just saying. 

BTW – These Butterfly research guys also physically characterized the BISIDE racket they used in the experiment.  Not too surprisingly, one of the few parameters they felt was important to record was the racket frequency. – Again, just sayin’

5.  > HEX talks plate theory but does not recognize that the plate theory equation he has seen assumes an isotropic and homogeneous material

I very much recognize this, so the issue is that my point hasn't been understood. To elaborate further, if what you're claiming about the relative significant of surface vs inner plies is true it should certainly show up in this category of theories as some relative weighted factor with respect to distance from center (then integrated over the thickness of the plate).

> So in estimating blade stiffness, can we only look at blade thickness h as HEX says?  No, you have to look at both E and h and the plate theory equation clearly shows this.

Really? Especially when you make the exact same point I do?:
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Yes really really Hex and I do it because I made this exact point much earlier in this thread, (originally in 2008 six years ahead of you) and that original point is:

Now because you cannot admit that you have read it, you’ve been claiming that I am ignoring parameters such as blade size (but blade size = A *L so that opinion of yours is now proven false) and blade mass, length, modulus E, area moment of inertia.  You keep repeating this, while willfully ignoring the fact that the parameters are in the equation.  You also willfully ignore that as I derived it so would I know they were there, and since I have written them down, that I have not ignored them.  Since you could not see the elephant in the room, I adopted your strategy.

I thought you would be upset if I adopted the same argument against you as you were making against me – a dose of your own medicine and overdue.  You ignored my results and then claimed that parts of it were yours.   You’ve just complaining that I have unfairly adopted your point that stiffness E*I and thickness matters.    Wow……..I am underwhelmed.    Yes, stiffness and size and mass and area moment of inertia and thickness are clearly in my equation above. Further, the relationship above is much more explicit and carefully thought through than the verbal pondering you have presented (of course you actually have not presented anything other than words).  

 SO…to recap, I am making the exact same point you have made, but its my right since I was the first to make it, and I did it in writing years before you showed up.   I can’t claim a win here since you were never actually in the point.  Of course all of this is simple engineering and well understood by many.

6. ) JRSDallas said "Finally, for any given blade construction, increasing thickness h very quickly makes it stiffer – i.e. stiffness grows as the third power of thickness (as originally posted 05/22/2013 at 7:24am, paragraph 6). "

6). Hex said:  “So it appear you too grasp the differing divergence of a linear vs 3rd-order polynomial function. In practice most TT blade woods don't differ much anyway: most surface species are ~10-13 GPa, and inner ones are in similar relative range.”    

Gee I guess so.   I prefer composites because of the wider design range they offer.  Recently I was surprised by the calculated effect of a hard wood outer ply overwhelming the stiffness contribution from carbon layer below it.   I could just say that all 7mm blades play the same, but I have had too many and I know they don’t.  

> As I remember, that formula also included structural terms that let us understand the connection between the mode frequencies and how the layers of the blade contributed. 

7.) Just because a function contains some of the right variables and seems to go in the right direction doesn't mean it's necessarily accurate. You might recall straight up stiffness contains the same variables without some of the drawbacks of frequency. You might also recall that a relative simple transform to any of these "kind of right" unbounded indicators make them significantly more right. In the real world wood density/hardness tends to vary and it's usually not possible to measure frequency before buying anyway, so in practice considering overall thickness with some compensation for any usual characteristic gets maybe 80% accurate results for very little effort. I seriously doubt more complicated metrics for standard blade is more than 10% better, and in any case probably less so than just pumping the rather trivial aforementioned transform if there are significant differences in thickness/rigidity.

There you go again talking about unbounded indicators.  WWWWOOOOOOUUUUU there must be some sort of deep spooky insight or dark magic danger from an unbound indicator akin to the role that prime numbers density plays in the density of quantum chaos energy levels or in Bessell function eigen modes – Oh my god frequency again!  Nu Nu Nu Nu….  Nu Nu Nu Nu….

It is up to the reader to apply some intelligence as to when a difference of frequency is relevant to their task of comparing the relative stiffness of two blades.   In every case, I have presented the use of frequency as a discriminator in those terms.  It is a very sensitive discriminator and a reasonable tool.  Further, I took pains to point out the variability in wood samples even when building many copies of the exact same blade.  

Yes, most of us know that blade to blade differences for even the same model are large so yes, buying any blade is a crap shoot even after you account for thickness, and compensating for unusual characteristics. Touching the blade and even better playing with it are the best indicators.   But…….don’t forget that the thread topic is to understand the playing characteristics of wood.    SO…..the goal should be to take the effort to actually try to understand rather than argue that it isn’t worth bothering to understand.  

Everyone can argue that they will never use some topic or other that they had to study in school.   I would argue that knowledge for knowledge’s sake creates both a rich internal life and improves the fabric through which we see the world.

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AgentHEX wrote: Please try to understand what is being said before going off. I did not make the statements above nor below without careful consideration and any possible error will not be trivial. OTOH the error in that frequency equation rather is. > Well since you have by omission left me with my choice of what frequency is correlated with, I hereby claim that frequency is directly correlated with frequency Your general assertion through this thread and reinforced just above is that frequency is a good first order approximation of speed. This is not in dispute.  Then why is there any further discussion?.  Its useful. OK subject closed.  Apparently not.  What is in contention is the quality of approximation and its exclusivity to frequency, namely that other simple "indicators" are similarly valid (and flawed in their individual ways). I don't believe you're disputing this, or at least haven't done so above. I agree that I have never said frequency is an "exclusive" indicator nor that no other “indicators” exist or are useful.  I also agree that you have adopted the language “what is in contention” in the paragraph above as an innuendo so you can argue that you have a greater perspective that separates you from a view that was never expressed.  You then close, by saying that you don’t believe I am disputing your position or at least haven’t done so.   You make a specious claim to establish a Potemkin controversy to look good against.  What is actually in contention is your position on insisting that better players should use your paddle so that it can learn from them how to play better, and on how race and religious preference are “indicators” of table tennis success – when other simple “indicators” like visual learning, cumulative training, and opportunity are similarly valid and less flawed by the influence of cultural and sampling frame biases.  I don’t believe you’re disputing this, or at least haven’t done so I would hope. > Your parenthetic statement (they certainly aren’t with respect to size, and who knows what else) says you claim that (frequency) is not correlated with blade size or other physical parameters.      given that statement since I claimed the exact opposite, and this claim is exactly the reason why frequency is particularly flawed as mentioned at least twice above: speed will not vary significantly with size, as better predicted by, say, stiffness rather than frequency. I noted this to punctuated these particular flaws. I purposefully misunderstood you with the thought that you might see that I was imitating you. > yes the contribution of plies that are displaced from the plywood mid-plane do correctly does show up as a distance squared weighting factor per the Parallel Axis theorem > Now you admitted as much earlier in this thread, you said when you did not understand how I was using the parallel axis theorem to calculate the area moment of inertia I of the laminate beam in the first place.  First of all, please consider that I was trying to be polite. In hindsight this was a poor idea. The error is actually pretty obvious and the reason it evaded capture only slightly less so. I've detailed above exactly why the use of Parallel Axis theorem here is wrong. More briefly, note that it is by definition calculated from the center of mass to the center of rotation. By your own assumptions for the beam equation, the center of rotation is around the "end" of the blade (ie the fixed neck point), which is certainly reasonable. But note that the distance from that point/axis to blade center of mass is quite a bit further and in any case perpendicular to the few mm the plies are shifted. Before being dismissive, draw out the diagram with both axis specified, it's not hard to see the problem and I would expect any first year physics student to be able to do it. The way Parallel Axis is being used in your freq equation is a fundamental error in the basic geometry involved. The specific numeric illustration a few posts back should provide additional illumination. The reason why this clear error is not evident (ie blows up the results) is that the terms involved are rather small as mentioned. Thank you for being explicitly clear so that I and others can understand. You are right that it is not hard to see the problem.   It is absolutely true that when a free body is rotating, its moment of area is calculated with respect to the principal axis of rotation and that it passes through the center of mass (COM) of the body.  Similarly, if using the Parallel Axis Theorem to account for a segment of that body, that distance should also be from the COM of the segment to the principal axis of rotation that passes through the COM of the body. However, vibration analysis solutions address the internal degrees of freedom of the body, and are independent of the 3 translation and 3 rotation degrees of motion freedom of the body.   Free vibration does not cause rotation around a rigid body’s center of mass, and our cantilever constraint is just a boundary condition on the differential equation for the free vibration of the beam.  Applying boundary conditions will shift the theta values of the modes which shifts the final frequencies but they don’t make the beam translate or rotate.        So….when looking at the vibration modes of a beam with uniform cross section (like ours), the moment of area is taken with respect to the center of mass of the beam's cross section.   For a beam with rectangular cross section, if it was a single material, its moment of area would be I= (bH^3)/12.  However since our beam is a symmetric multi-layer stack of rectangles (of different material), and having correctly applied the Parallel Axis Theorem, the correct moment for each layer is as I previously presented.  In conclusion then, the effect of plywood layer stiffness and placement within the plywood stack as described by the equation is correct, and the size of the effect are exactly are originally presented.  A page from Stokey Chapter 7 Vibration of Systems Having Distributed Mass and Elasticity shows that I have used the correct axis of rotation and have correctly calculated the moment for a cantilever.  By extension, this means my equation is correct and that the effects I calculated based on layer placement are correct.   However, also as mentioned it's visible how this came to be in the first place: you're trying to account for the intuition that the outer plies have more impact than inner ones, and way how parallel axis seems to work out (with its wrong application) assuages that concern since bigger numbers are multiplied for plies further out. This intuition is not in dispute given you just repeated it in another post above. The weighing is however completely wrong and certainly not the weighing in actually plate equations I was referring to. To tie this all together, consider a blade with a hard surface on only one side (and the other missing or whatever). Note your equation is agnostic to which side the impact is on; surely they differ if surface layers matter, which rather ruins that "greater surface effect" illusion the equation is supposed to illustrate.   Of course it is agnostic, it solves for a symmetrically laminate beam.   I can extend it but I don’t need to prove anything here. Again, to provide some perspective, a careful particle sim of the whole collision would probably show some surface effect, but this has to do with the peculiarities of how internals of solid materials work, and nothing to do with rotational inertia. Analysis of such matters are well beyond what can be provided by simple clean equations. You were very careful in explaining your objection to my moment of area calculation on the beam.  I carefully answered and I provided 3rd party support to corroborate my position, I assume that you now accept my calculation.   My equation describes how the plywood construction leads to its vibration frequencies and how selecting what materials and where in the stack they are placed creates a calculable effective stiffness.   In particular, my statement that moving the stiffest plies to the outside has the greatest increase on moment and therefor on system stiffness and therefor on frequency as per the equation. You have coined concept you call “A GREATER SURFACE EFFECT” and are using it in regards to the collision of a ball on that surface but this has nothing to do with your objection to the equation I derived.   You are mixing apples and oranges.   I agree that oranges are interesting but you not having an orange does not mean I do not have an apple.   Now you recommend a careful particle sim be performed of the total collision.  I agree, please perform the careful sim you recommend and get your orange.  Show us your results because it is an interesting topic. I don't have access to the tools to do a collision FEA with sufficient node count to be useful.    My work in this thread has been on the engineering properties of woods, selection of woods, the engineering of plywood and vibration modes of a cantilevered plywood, how plywood design effect plywood vibration frequencies, example calculations of multiple different plywood designs, FEA modeling of a single ply table tennis blade (Amultart dimensions), measured sound spectrums from struck similar size blades of differing plywood constructions, and qualitative correlation between frequency and speed feel of these blades.  Energy loss considerations support a connection with increased COR but no quantitative measurement of COR, nor full FEA collision modeling have been performed. Reader’s should note that I do not agree or disagree with the conclusions Hex is espousing regarding the magnitude of blade or rubber selection has.  For me, these issues are outside the topic of this thread and I myself have not worked through them to the point that I can prove a clarifying insight –  There is obvious merit to Hex’s saying it’s the player and not the equipment –Yep. At the same time there is obvious merit in saying that players do feel the differences in equipment-Yep.  And finally there is merit in noting the obvious that in all sports, that as the best players are very careful in selecting their equipment, there may be causal reasons for doing so – Yep.         I look forward to seeing any efforts that can replace the arguments about what also floats on water besides witches. > Please also refrain from using Wolfram Alpha to pull up some non-physical function (as you did earlier in this thread) to argue that it is a better predictor.  We all know that the declining number of pirates is strongly negatively correlated with the increase in global temperatures.  I'm not sure how this is supposed to make any sense. My Wolfram Alpha plot is just how any realistic speed vs (thickness, freq, etc) plot must necessary look like. If that's not clear, please consider what bounded vs unbounded means again. Any unbounded method of determining it is necessarily wrong without some attempt to massage the math into a sane shape. This is basic physical modeling 101. So when I followed your link it did not lead to a plot, just an equation with no discussion of causal parameters.  If you had Wolfram Alpha derive that equation based on a physical model you set up and then have it create a plot then please walk us through those details by dropping screen captures into a post.  Show your work.        Still I believe you made this argument to show that my equation was not physical or something.  I've shown my equation was properly generated so this contention is now irrelevant.

Wrap up:

I invite and look forward to seeing any work from anyone on the topic of collision.

As for myself, I’m going to move on now and spend more of my vacation time with my wife. 

AgentHEX wrote: (...)  With a TT blade that rotational axis is nearer the end of the blade (where you presumably hold it) and the line to its center is along the length of the blade. But JRS is calculating that line from the center outward towards the face. From first principles when you move the object (plies in this case) in same direction as the one the object swings in, it doesn't really change inertia at all which so the correction was unnecessary. (...)

Calculating the distance from the center towards the face is crucial for measuring the stiffness of the blade and I am surprised that you seem to ignore this fact - properties of such composite constructions are being deeply studied in modern technology and manufacturing in things like rocket fuel tanks, super-light sport yachts and even cars bodies. If you mean that these calculations are wrong then please say exactly where the mistake is, because so far you are only presenting puzzles to readers and giving some homework to OP and personally I wouldn't even expect to get a serious answer for such nonconstructive critique.