Numerical Analysis of table tennis blades

hipnotic · Sep 7, 2018

Being an over thinker as i am, i wanted to see if i could predict the rating of a blade before i build it. So, i decided to make a numerical model of a blade and analyse the results by changing several properties. The shape of the blade and the mechanical properties of the materials are always the same, the thickness, direction of plies, arrangement of plies and length of the handle are variable.

We all know that wood is all different, even in the same species and even the same board. The mechanical properties are very hard to predict and, because wood is orthotropic, the properties are different in all directions. Also, this model doesn't account to several other factors as the type of glue used. Nonetheless, if the mechanical properties used are not changed, i can at least make a comparative analysis between the models and predict if a blade is going to be faster then the other.

To make this comparison i studied several factors:

The 1st mode of vibration - Related to the bending of the blade. It can give some idea if the blade is going to flexible.
The 6th mode of vibration - The membrane mode (3rd bending). This is what you hear when you bounce the ball on the blade. There are several measurements of this mode for various blades, so i can compare them with my predictions and establish a rating.
The displacement on the top of the blade for a load of 1 N applied in the center of the face. This is also related to bending and gives me an idea of the flex.
The predicted weight.

I will show some images of the model and the modes of vibration. The shape of the modes are the same for all the models, but the frequency changes.

Mode 1

Mode 6

hipnotic · Sep 7, 2018

I tried to model blades with the materials i sourced, wich are, Balsa, Englemann Spruce and Red Cedar for the cores and Sycamore Maple and Sapele for the outer plies. I will try to build a few blades from this list, maybe one of each rating, and compare them to my predictions. In the next table are the mechanical properties i used, i couldn't find all the properties for some common woods but the most important are the Young Modulus along the grain and the density.

	-	Limba	Ayous	Balsa	Kiri	Koto	Sap	Pine	Red Cedar	Eng Spruce	Syc Maple
Other names		Korina	Abachi, Samba		Paulowonia
Density (kg/m3)	0	550	380	140	305	590	670	510	380	385	615
Monnin Hardness		2.4	1.1	0.3		2.5	4.2		1.1	2.2
Compressive Strength (MPa)		47	30	11	21	54	60	42	31.4	31.5	55
Bending Strength (MPa)		86	52	20	38	105	110	79	51.7	62.2	98.1
Janka (N)		670	430	400	1335	4185	6280	3160	1560	1740	4680
El - E1 (MPa)		10490	7260	3710	4380	12080	13960	10060	7660	9440	9920
Et - E2 (MPa)				55.65			698		421.3	556.96	644.8
Er - E3 (MPa)				170.66			1549.56		6204.6	1208.32	1309.44
Glr - G13 (MPa)				200.34			1228.48		666.42	1170.56	1101.12
Glt - G12 (MPa)				137.27			823.64		658.76	1132.8	624.96
Grt - G23 (MPa)				18.55			293.16		38.3	94.4	198.4
Ulr - U13				0.229			0.297		0.378	0.422	0.424
Ult - U12				0.488			0.641		0.296	0.462	0.476
Urt - U32				0.665			0.604		0.484	0.53	0.774

In the next table are the results. The weight is only of the wood plies, so you must account for glue and handle.
Balsa cores are giving some weird results, i think it's because the wood is so light.

Xaoc · Sep 7, 2018

I was wondering when some1 would take blade analysis to a FEA software, but given all the variables even between wood from the same type - humidity, fiber orientation I thought the whole thing needed a lot of simplification to draw any conclusions.
I am really eager to read more about your project!

Is there any specific reason for the irregular spacing of your mesh?

hipnotic · Sep 7, 2018

The mesh follows a grid of 7,5x7,5mm, i didn't want triangular shells but it is impossible given the geometry. It is just sub-meshed were needed.

In the meantime i detected a major mistake. My shell local axes were not oriented correctly so all of the models are rotated 90º. The 0º orientation was in fact perpendicular

Also, the software is giving me some very weird result when i use orthotropic materials, i don't know why. No way i would get an off+ blade using those materials and thickness.

I guess i can use isotropic materials and disregard the perpendicular stiffness. In the outer layers that shouldn't matter much because they are thin and the perpendicular stiffness is very small compared to the longitudinal stiffness. However, the core is thicker so i will have results lower than expected.

tropical · Sep 8, 2018

Did you model with membrane elements or brick?

hipnotic · Sep 8, 2018

I used layered shell elements in SAP2000.

bobpuls · Sep 8, 2018

From what i now is balsa 300n janka hardnes.... One source from more
https://www.wood-database.com/balsa/
And ayous is definitely lighter then red cedar ,from my own experience....weird measurement...

tropical · Sep 8, 2018

hipnotic said:
I used layered shell elements in SAP2000.

I think for more accurate results and less surprises you may be better off with brick elements. Isotropic is a bit simplistic for porous media like woods and also glue could be used in the model as shell elements.

hipnotic · Sep 9, 2018

bobpuls said:
From what i now is balsa 300n janka hardnes.... One source from more
https://www.wood-database.com/balsa/
And ayous is definitely lighter then red cedar ,from my own experience....weird measurement...

Janka hardness is not importante for FEA. As for the weight of Ayous, you are probably right but i got those numbers from wood database, the tt blades database and other sources.

hipnotic · Sep 9, 2018

tropical said:
I think for more accurate results and less surprises you may be better off with brick elements. Isotropic is a bit simplistic for porous media like woods and also glue could be used in the model as shell elements.

I will try that. I used layered elements first because i wanted a simple way to change the parameters, with solid elements is a little bit more time consuming but i think the results will be better.

tropical · Sep 9, 2018

hipnotic said:
I will try that. I used layered elements first because i wanted a simple way to change the parameters, with solid elements is a little bit more time consuming but i think the results will be better.

And I think you can cut the model in 1/2 due to symmetry. Oh how I miss the time when I used to do earthquake analysis with Ansys (20 years ago...).

yogi_bear · Sep 9, 2018

I wish there is also a speedtest for bare blades measuring their speed when hitting the ball.

hipnotic · Sep 10, 2018

yogi_bear said:
I wish there is also a speedtest for bare blades measuring their speed when hitting the ball.

Yogi_bear, there a simple test you can make, just google "table tennis blade frequency". Basically you download a spectrum analyzer app, bounce the ball on the blade and see the frequency peak. There is a list of recorded frequencies for many blades. I'm basing my ratings on those findings:

Under 1,000 is DEF
1000 - 1,100 ALL
1,100 - 1,250 ALL+
1,250 - 1,350 OFF-
1,350 - 1,550 OFF
1,550-. OFF+

I think this is actually a good method. Higher frequency means the blade is stiffer, so the Young modulus is larger. A stiffer blade equals a faster blade, it's not the only variable but it's directly proportional.

yogi_bear · Sep 10, 2018

yeah I have seen that but I am more on the practical side of measuring speed though the amount of blades you would need is not practical.

hipnotic · Sep 11, 2018

I have made some progress on this. I refined the mesh and sorted out the weird results i was getting with balsa cores. Just a few notes about the modelling assumptions:

I'm using isotropic materials. I have found that the perpendicular stiffness is almost negligible, and using orthotropic materials produces inconsistent results.
I'm using layered shell elements. I tried modelling with solid elements but found that it was much more time consuming and the results were inconsistent.
I'm using the properties from the table for the materials i don't have. For the materials i do own, i can calculate the density and calibrate the model by adjusting the E.

This is an image of the new mesh.

I tried modelling one of the blades i've built.

With the materials from the table i got 1384 Hz, 83,4g without handles. That gives me a Off rating.
With the calibrated model i get 1308.2 Hz, 68g without handles. That gives me an Off- rating.
The blade came out at 73.4g without handles, with 1250 Hz in the bounce test. I consider this a pretty good approximation, the weight difference is mostly glue, which the model doesn't account for, but the rating is in the Off- range as expected.

Now i plan to make a parametric analysis for 5 ply blades. For example, maintaining the total thickness of the blade but changing the thickness of the core and inner plies. I can also keep the thickness constant and just change the outer plies by gradually harder materials. There are so many options...

brokenball · Sep 11, 2018

I am not convinced. It requires energy to make the paddle vibrate. That is energy that isn't returned to the ball. Doesn't the amplitude of the vibration have something to do with this? Higher frequencies and amplitudes indicate more energy in the vibration that isn't returned to the ball.

If the blade didn't absorb any energy at all it wouldn't flex or vibrate.

hipnotic · Sep 11, 2018

In a lab study you would need an external excitation to measure the natural response. In FEA the program excites the system "virtually". Also, i'm not getting in to the ball-blade interaction. I'm just calculating the fundamental modes of the blade and estimating the speed by looking at the 6th mode. Higher frequency = stiffer blade = more speed. This frequency is simply measured by the bounce test so i can compare to the FEA results.

tropical · Sep 11, 2018

True! Fundamental frequencies are independent of external excitation and FAE relies on boundary conditions to find them. With modal analysis people use external forces to find out these frequencies.

f= √(k / m) ÷ 2π is independent of force! (first fundamental of a simple harmonic oscillator )

But I still think your models are a bit off and not useful to compare with bounce test. FAE is easy to tweak and needs to be more detailed as you will find out. Simplifying model is ok as first lesson only.

hipnotic · Sep 11, 2018

tropical said:
True! Fundamental frequencies are independent of external excitation and FAE relies on boundary conditions to find them. With modal analysis people use external forces to find out these frequencies.

f= √(k / m) ÷ 2π is independent of force! (first fundamental of a simple harmonic oscillator )

But I still think your models are a bit off and not useful to compare with bounce test. FAE is easy to tweak and needs to be more detailed as you will find out. Simplifying model is ok as first lesson only.

You might be right but i'm only looking for a simple way to predict the speed of a blade before i build it. And to understand what will happen if i increase the core thickness and reduce the other layers, things like that. So far i have modeled one blade i built and a few that i know the composition, like the nittaku acoustic, with good results.

tropical · Sep 11, 2018

Core thickness and other layers thickness can be easily changed with parametric modeling; i.e. define t1=x, t2=xx, etc.. Also define dx=tx/n to control the mesh on thickness if you like. Nowadays FEA is so easy to use that users normally forget how to set up to do multiple what-if's with parametric and a touch of button. Good luck..

Numerical Analysis of table tennis blades

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