Numerical Analysis of table tennis blades

brokenball · Sep 11, 2018

Hipnotic, anybody, show us an equation where frequency of vibration is used to calculate the speed after contact.

hipnotic · Sep 11, 2018

brokenball said:
Hipnotic, anybody, show us an equation where frequency of vibration is used to calculate the speed after contact.

I think you are missing the point, the natural modes and their frequency only tell you the properties of the blade. If you excite the blade with that frequency, it will start vibrating in the shape of the corresponding mode, just like a bridge in the wind. I'm not calculating the speed, i'm just comparing them to the speed of existing blades by finding their natural frequencies. Also, again, it's not meant to be exact, i just want to have an estimation.

tropical · Sep 12, 2018

Imagine a guitar with 6 strings : Each string has its own natural freq, see the table below

As you can see the higher the nat freq of a spring the higher the note we can hear. Same analogy to the blade the higher the nat freq the stiffer it is or , thus is how DEF, ALL, OFF is used for certain range of nat freq of the blade.

Velocity of a spring is in the relationship of this formula: frequency = speed/wavelength

So speed = Fn*lamda (wavelength) where in a simple spring lamda for 1st mode of vibration is 2L (L is the spring length)

Hope this helps...

brokenball · Sep 12, 2018

tropical said:
Velocity of a spring is in the relationship of this formula: frequency = speed/wavelength

WTF?
That formula makes sense for sound waves or light waves traveling through air or space.
A spring does not move like that. It vibrates
pos(t)=Amplitude*sin(ω*t)
vel(t)=Amplitude*cos(ω*t)*ω
acc(t)=Amplitude*-sin(ω*t)*ω^2
Where ω=sqrt(k/m) as you pointed out earlier.
This doesn't take into account the damping or attenuation of the vibration.

However, this is not getting closer to answering my question. How does the blade's frequency of vibration affect the speed of the paddle? I still haven't seen a formula.

What is the amplitude of vibration? If the amplitude is small, does it really make a difference? From the formula for velocity above, the amplitude and frequency both contribute to the velocity of the vibration.

What we really need to know is how the ball, rubber and blade interact.

tropical · Sep 12, 2018

What we really need to know is how the ball, rubber and blade interact.

This sentence shows you completely ignore the OP as the OP was analyzing the blade only. What the hell we want to use FAE with ball, blade, rubber altogether? Come on!

I think you need to polish your basic understanding of vibration as it is all about fundamental freq'es like we discussed. I was trying to explain the basics only and you seem to get offended and was trying to make matter more complex by adding damping in the equation. I am not sure you understand what we are talking about here.

What we may need to argue about is whether the premise that DEF, OFF, ALL with fundamental frequencies are acceptable or not. The OP was trying to use FAE to match previous experiments and draw conclusion.

I feel you lack understanding of basic vibration thus feel helpless here as I can't help you further. Good luck with your search ....

hipnotic · Sep 12, 2018

Ok, i have another approach. First let me remind whoever is reading that this is just an attempt to create a simple way of comparing blade speed. It's not meant to be exact and definitive. Blade speed is not a number, it's a range, it varies depending on several factors, too many to be calculated.

Some basic concepts:

We all know a stiffer material will produce a higher bounce. For example, if you bounce a ball on a 10 mm metal sheet and on a 10 mm piece of cardboard it will bounce higher on the metal sheet, right? Higher bounce means the ball will leave the metal sheet faster, so a stiffer material will produce a faster rebound.
Each layer of a blade has its own Young Modulus (E; stiffness). The stiffness of a blade depends on the arrangement of the plies and their distance to the core. The further the distance the E becomes higher.
As a result, a homogenized E can be calculated by knowing the the stiffness, thickness and distance to the core of the ply. This homogenized section would produce the same deflection has the real blade when subjected to the same load.
Frequency is dependent on mass and stiffness. If you increase the stiffness the frequency will increase, if you increase the mass the frequency will decrease, and vice-versa.

So, in order to compare the frequency we have to take in to account both the mass and the thickness of the blade. That's why balsa blades have such high frequencies, because their mass is lower, but the homogenized E will also be low unless you use composite materials or thick cores.

To study this i made several calculations which are presented in the table below. For analysis purposes i just have 5.7 and 5.8 mm sections, but i plan to make more in the future. For each thickness there are 3 weights (these are weights without handles), and for each weight is the frequency for various homogenized sections ranging from 3000-7000 MPa. I have found that most blades are within this range.

	5.7 mm	5.8 mm
Weight (g)	60	65	70	60	65	70
Density (g/cm3)	0.458	0.496	0.534	0.450	0.487	0.525
E3000	1043.6	1002.9	966.5	1071.4	1029.9	991.9
E4000	1205.1	1158.0	1116.1	1237.1	1189.2	1145.3
E5000	1347.4	1294.7	1247.8	1383.1	1329.5	1280.5
E6000	1475.9	1418.3	1366.9	1515.1	1456.4	1402.7
E7000	1594.2	1531.9	1476.4	1636.5	1573.1	1515.1

For example, if a blade is 5.7 mm weighs 85g we subtract about 20g for handles so we get 65g. Imagine we get a reading of 1250 Hz in the bounce test, that would be equivalent to a stiffness of about 4672 MPa (by interpolation). Now imagine a second example of a blade with 5.8mm, 90g and 1250 Hz. That would give a stiffness of about 4778 MPa. So, by comparison, blade two would be a little faster even though they have the same frequency in the bounce test.

In conclusion, i think we can establish speed ratings based on stiffness, by finding the homogenized E of several well known blades in each rating.

brokenball · Sep 12, 2018

tropical said:
I think you need to polish your basic understanding of vibration as it is all about fundamental freq'es like we discussed. I was trying to explain the basics only and you seem to get offended and was trying to make matter more complex by adding damping in the equation. I am not sure you understand what we are talking about here.

My knowledge of underdamped systems is perfect. I work with them all the time. I can even use actual vibration/motion data to create models.
This is something the OP will need to do to verify his FEA otherwise the model will not be very good a predicting anything.
If the OP just wants to simulate vibrations in a blade then OK but correlating the model to def, def+ … off+ ) will be a challenge.

hipnotic · Sep 12, 2018

brokenball said:
My knowledge of underdamped systems is perfect. I work with them all the time. I can even use actual vibration/motion data to create models.
This is something the OP will need to do to verify his FEA otherwise the model will not be very good a predicting anything.
If the OP just wants to simulate vibrations in a blade then OK but correlating the model to def, def+ … off+ ) will be a challenge.

You are still missing the point. I'm not correlating the model to speed ratings, i'm correlating the model to existing blades which have established speed ratings and stating if they are faster or slower. And it's not a challenge, it's presented in the table above.

tropical · Sep 12, 2018

Hey hipnotic ... did you see this thread? It is very interesting topic that might help your modeling.

http://mytabletennis.net/forum/foru...tle=blade-performance-vs-wood-type-and-design

hipnotic · Sep 12, 2018

tropical said:
Hey hipnotic ... did you see this thread? It is very interesting topic that might help your modeling.

http://mytabletennis.net/forum/foru...tle=blade-performance-vs-wood-type-and-design

Thanks Tropical. Yes, i saw that thread, in fact it's what gave me the idea to try my own analysis.

For those who doubt, or don't understand, that there is a correlation between the bounce test, frequency and blade speed i suggest a couple of papers:

https://www.sciencedirect.com/science/article/pii/S1877705812017316

https://www.sciencedirect.com/science/article/pii/S187770581201716X

langel · Sep 12, 2018

I can't see in this papers any correlation between frequency and speed explained.

hipnotic · Sep 12, 2018

langel said:
I can't see in this papers any correlation between frequency and speed explained.

That is commom sense and i have already explained a few times along the topic: Stiffer=higher bounce=faster

langel · Sep 13, 2018

The question about the common sense is clear.
Its clear that there is some corelation between the frequency response and the speed capabilities of the blade.
But what I say is that in this papers, due to the limited scenario of the tests, this corelation is not explained in details needed to obtaine trusted results.
The tests are executed with only one ball speed - 5 meters per second, which is in the lower end of actual playing speed.
Blades with different composition may have different frequency pitch at that speed, never mind what is their actual playing speed range. I have such blades - some have very low pitch at lower speed impact, but are actually OFF+, some have higher pitch at lower speed, but actually are slower OFF- max.
My opinion is that the actual speed range depends much more on the particular blade structure and many other factors depending on this structure, which combined in a system, produce very difficult to analyze rezults with a limited simple test.
I would like to see a test with different composition, different flex and executed with different ball speeds from 5 meters per second to 30 meters per second with many steps, measuring the rebounce speed too. Such a graph would give much more trusted results.
As I said above a single low speed ptch test may lead to wrong conclusions.

hipnotic · Sep 13, 2018

Ok, forget the frequency. Frequency is just the way i have to find the Young Modulus of the existing blades. Without knowing their composition all i have left it's the measured frequency, thickness and mass. With that i can calculate the approximate stiffness of the homogenized section based on the table i present here. The ratings i attributed were based on the analysis of a few blades so they are not definitive.

			5.2 mm	5.6 mm	5.7 mm	5.8 mm	6.2 mm
		Weight (g)	60	65	70	55	60	65	70	60	65	70	60	65	70	45	60	65	70
		Density (g/cm3)	0.502	0.543	0.585	0.427	0.466	0.505	0.543	0.458	0.496	0.534	0.450	0.487	0.525	0.316	0.421	0.456	0.491
E (MPa)	All	3000	909.3	874.3	842.3	1061.9	1016.5	976.5	941.7	1043.6	1002.9	966.5	1071.4	1029.9	991.9	1366.7	1184.0	1137.7	1096.4
All +	4000	1049.9	1009.5	972.6	1226.2	1173.8	1127.5	1087.4	1205.1	1158.0	1116.1	1237.1	1189.2	1145.3	1578.1	1367.2	1313.7	1266.0
Off -	5000	1173.8	1128.6	1087.3	1370.9	1312.3	1260.6	1215.7	1347.4	1294.7	1247.8	1383.1	1329.5	1280.5	1764.3	1528.6	1468.7	1415.4
Off	6000	1285.8	1236.3	1191.1	1501.8	1437.5	1380.9	1331.7	1475.9	1418.3	1366.9	1515.1	1456.4	1402.7	1932.8	1674.5	1608.9	1550.5
Off +	7000	1388.8	1335.4	1286.5	1622.1	1552.7	1491.6	1438.4	1594.2	1531.9	1476.4	1636.5	1573.1	1515.1	2087.6	1808.6	1737.8	1674.8

In the models i'm not even looking at the modes, i just look at the displacement for the applied load. Then i use the table below to calculate the Young Modulus of the homogenized section, so i can compare to the stiffness of the existing blades.

	e (mm)	6.2	6.1	6.0	5.9	5.8	5.7
E (MPa)	1000	0.1728	0.1814	0.1906	0.2005	0.211	0.2223
2000	0.0864	0.0907	0.0953	0.1002	0.1055	0.1112
3000	0.0576	0.0605	0.0635	0.0668	0.0703	0.0741
4000	0.0432	0.0453	0.0477	0.0501	0.0528	0.0556
5000	0.0346	0.0363	0.0381	0.0401	0.0422	0.0445
6000	0.0288	0.0302	0.0318	0.0334	0.0352	0.0371
7000	0.0247	0.0259	0.0272	0.0286	0.0301	0.0318
8000	0.0216	0.0227	0.0238	0.0251	0.0264	0.0278
9000	0.0192	0.0202	0.0212	0.0223	0.0234	0.0247
10000	0.0173	0.0181	0.0191	0.02	0.0211	0.0222

langel · Sep 13, 2018

Ok, so its a method to analyze and compare the stiffness.
Still I have some rematks, that in many cases the general common sense about the corelation between stiffness and speed range may not work as expected.
Anyway every research in that area should be welcome and respected.

hipnotic · Sep 13, 2018

langel said:
Ok, so its a method to analyze and compare the stiffness.
Still I have some rematks, that in many cases the general common sense about the corelation between stiffness and speed range may not work as expected.
Anyway every research in that area should be welcome and respected.

Balsa cores are a little tricky because the stiffness/weight ratio is high and the mass is low. Right now i'm only analyzing 5 ply all wood blades, and getting good results, but if i introduce composites then the problem gets more complex. Also, there are stiffer glues then others, and sometimes they play a major role in the performance of the blade.

langel · Sep 13, 2018

hipnotic said:
Balsa cores are a little tricky because the stiffness/weight ratio is high and the mass is low. Right now i'm only analyzing 5 ply all wood blades, and getting good results, but if i introduce composites then the problem gets more complex. Also, there are stiffer glues then others, and sometimes they play a major role in the performance of the blade.

Yes, the more complex the structure, the more sofisticated the research will be.

zeio · Sep 13, 2018

Just throwing it out there.

1st Mode of Vibration (Hz)
~~A certain~~ PLStar 5-ply all-wood blade - 39.58±0.86
~~A certain~~ Keyshot Light 3+2 Vectran blade - 39.97±1.21
~~A certain~~ Primorac Carbon 3+2 Carbon blade - 54.56±0.49

Coefficient of Restitution in the Normal Direction at Low/Medium/High Impact Velocity (7.41±0.4/8.28±0.52/9.16±0.46 m/s)
Blade Alone
5-ply all-wood - 0.7±0.01, 0.7±0.04, 0.73±0.04
3+2 Vectran - 0.7±0.03, 0.7±0.03, 0.71±0.02
3+2 Carbon - 0.75±0.05, 0.75±0.05, 0.79±0.06
With Sriver in 2.0 On Side of Impact
5-ply all-wood - 0.92, 0.86, 0.84
3+2 Vectran - 0.92, 0.87, 0.84
3+2 Carbon - 0.93, 0.89, 0.86

Contact Duration (ms) at Low/Medium/High Impact Velocity (7.41±0.4/8.28±0.52/9.16±0.46 m/s)
Blade Alone
5-ply all-wood - 1.74±0.17, 1.53±0.10, 1.40±0.07
3+2 Vectran - 1.68±0.08, 1.47±0.06, 1.38±0.21
3+2 Carbon - 1.44±0.23, 1.40±0.07, 1.32±0.05
With Sriver in 2.0 On Side of Impact
5-ply all-wood - 1.67, 1.47, 1.33
3+2 Vectran - 1.67, 1.43, 1.27
3+2 Carbon - 1.37, 1.2, 1.1

BTW, the blades are rated medium, medium-fast, and fast by the manufacturer.

Feel free to thank me by filling up my piggy bank.

2023/8/5 Added blade model and dwell time

tropical · Sep 13, 2018

langel said:
The question about the common sense is clear.
Its clear that there is some corelation between the frequency response and the speed capabilities of the blade.
But what I say is that in this papers, due to the limited scenario of the tests, this corelation is not explained in details needed to obtaine trusted results.
The tests are executed with only one ball speed - 5 meters per second, which is in the lower end of actual playing speed.
Blades with different composition may have different frequency pitch at that speed, never mind what is their actual playing speed range. I have such blades - some have very low pitch at lower speed impact, but are actually OFF+, some have higher pitch at lower speed, but actually are slower OFF- max.
My opinion is that the actual speed range depends much more on the particular blade structure and many other factors depending on this structure, which combined in a system, produce very difficult to analyze rezults with a limited simple test.
I would like to see a test with different composition, different flex and executed with different ball speeds from 5 meters per second to 30 meters per second with many steps, measuring the rebounce speed too. Such a graph would give much more trusted results.
As I said above a single low speed ptch test may lead to wrong conclusions.

Frequency (peak) is easiest to measure with an app (spectrum analyzer) so many testing have been done and most give very good correlation between peak noise and rating (OFF, ALL, etc). There might be some exception but very rare per the database below.

https://docs.google.com/spreadsheets/d/1tnzuhP98Iwl3_ZYIKs770Z4GeEXB1cPaF6xXC3IMLfg/edit#gid=0

If you have a blade that seems to be out of norm you can take a look at the table and find out. For me all 3 wood blades I currently have are right on!

langel · Sep 14, 2018

tropical said:
Frequency (peak) is easiest to measure with an app (spectrum analyzer) so many testing have been done and most give very good correlation between peak noise and rating (OFF, ALL, etc). There might be some exception but very rare per the database below.

https://docs.google.com/spreadsheets/d/1tnzuhP98Iwl3_ZYIKs770Z4GeEXB1cPaF6xXC3IMLfg/edit#gid=0

If you have a blade that seems to be out of norm you can take a look at the table and find out. For me all 3 wood blades I currently have are right on!

Ok, I don't want to argue and I'm not doing it at all.
Just want to clarify some points - I'm very familiar with these tables and we have discussed the pitch tests many times in this forum. All these tests are held by many people from all over the world. But they all are executed by simple drop of the ball. And yes, there is a corelation, though some results vary a lot. As these tests are executed with a very low ball speed, they would be more precise with slower blades, blades with simpler composition and blades with tighter range - no matter def, all or off but tighter in performance. The more complicated the structure of the blade is and/or the more wide is its speed range, the more unaccurate or even false such a test may be. An example - the 11 ply Palio v1 /7 wood, 4 carbon/ has a low-speed ball pitch like a slower 5 ply DEF wood, but in fact is OFF, even OFF+. Palio TNT-1, 7+2, is mine stiffest and hardest OFF+, even OFF++, but because is massive and thick its pitch is like a OFF-. Rosi Emotion sounds like OFF, but is All+, OFF- max. All mine Xiom Vega Tour blades have higher pitch than those in the tables, but still I can't say are they faster - it may be just because they are of lower weight. Besides that some blades are more sensitive to the balls and with different balls they sound different - it may be because of the outer ply, overall composition and structure, etc.
So I don't say that there is no corelation at all, a pitch test may be somewhat indicative, but has to be trusted with care. For me it would be useble if you compare to chose a blade between several pieces of same model.

Numerical Analysis of table tennis blades

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