Making sense of modal analysis 3

[rfulky]

Say you did modal testing on a baseball bat and determined the 1st 15 modes via FEA. Then you did impact testing on the same bat at several impact forces that could be produced by a human. How do you find what force excites what modes and does every force have to excite any modes? Also is their any information about what frequencies a human can detect in their hands? (ex. a human hitting a base ball and it feels good or a human hitting a baseball and it stings their hands)

 

[pjhype]

Modal analysis (FEA) and modal (impact) testing should give you identical results (up to a certain frequency) provided that the boundary conditions in your model match the boundary conditions in your test. The only exception would be if in your experiment you were to excite the structure at a nodal point for a given mode and not excite that mode.

For linear systems, it is usually not necessary to excite the structure with forces that are of the same magnitude as the operational force, since you are only interested in obtaining transfer functions in modal testing. That is, the displacement (acceleration) response is normalized to the magnitude of the excitation (impulse) force.

The quality of a hit is usually a subjective measure that involves a few observations: (1) the absence of stinging in the hands, (2) the sound of the ball hitting the bat, and (3) the distance that the ball travels. Usually, there is one region on the bat (the "sweet spot&quot that gives you the optimum hit. (There are probably more subjective factors that determine the optimum hit, such as the balance of the bat during impact of the ball, damping (lack of ringing) in the bat, etc.)

I think that what you will want to do is find the point on the bat that you could hit the ball that will give you the maximum spring back on the ball and will result in minimal movement at the bat handle (a nodal point). This process can involve the following:

You will need to perform a finite element modal transient analysis where you might fix the bat at the handle (a first order approximation) and based on momentum exchange between the ball and the bat, apply an impulse to the bat at various locations along the length. You might try to vary the impulse time to account for differences in ball hardness, bat speed, etc. Your output will be the constraint forces at the handle as a function of time. What and how you minimize the constraint forces could involve a look at several factors, including peak overshoot, frequency, logarithmic decrement, etc.

These types of analyses have been performed extensively in the golf club industry and the tennis racquet industry, and the methodology will be almost identical for baseball bats. You might find some published papers that go into a lot more detail on this subject.

pj

 

[GregLocock]

Each mode is excited by an impact unless the mode is nodal at the point of contact. The proportion of the force that goes into each mode is found by using modal superposition, although that is a bit circular since in practice th frequency content of the impact will be governed by the mobility at the contact point. Hmm, don't really know the answer.

Every force has to excite some modes, yes (not necessarily in your first 15 tho).

The human hand is responsive to vibrations up to at least 200 hz, your situation is analagous to many studies that have been done on manual workers, so you should be able to find some papers on it. I doubt you'll find much on the web, but generally the topic is "Human response to vibration". B&K have published some useful books on this, and Mike Griffin at ISVR at Southhampton University in the UK has definitely published some good stuff.

Cheers

Greg Locock

 

[hacksaw]

As stated by Greg, the response is a summation of modes. the excitation of each is determined methods covered in standard texts.

 

[MikeyP]

I came across this site while looking for papers on the dynamics of bowling balls!

Google works in mysterious ways.

http://www.gmi.edu/~drussell/Demos/bats/bats.html

M

 

[MikeyP]

What the above article says is that the first few modes of the bat have nodal lines at roughly the same point. This is the "sweet spot". A ball impacting at this point does not excite many modes. As a consequence less of the ball's energy is transferred to vibrations in the bat. This has two effects: 1 The ball bounces off the bat faster and 2 the hitter feels less jarring and vibration as there is less vibrational energy transferred to his/her arm.

If you have performed a modal analysis on the bat then it is relatively straightforward to calculate which modes will be excited and to what degree. The maths behind it can be found in modal analyis texts (eg D Ewins "Modal testing: theory and practice&quot. It involves some very basic matrix maths, and is therefore difficult to write in a post here but in words...

From the modal analysis you get information on Natural frequency, modal mass, damping ratio and mode shape for each mode you have tested. These 4 parameters completely describe the structural dynamics up to the frequency of the highest mode you have tested (under the assumption that the structure is linear and that damping is "proportional", both reasonable assumptions).

From these parameters there are simple relationships which will give you the modal damping and modal stiffness for each mode. Arranged into diagonal matrices, these form the modal mass, modal stiffness and modal damping matrices. These are directly analogous to the mass, stiffness and damping matrices in physical space and we can calculate a frequency response matrix in exactly the same way...

H = 1/( -M*omega^2 + C*j*omega + K )

This is the FRF matrix in "modal space". The FRF matrix in physical space tells us how much response we get at point x if we apply a force at point y. The MODAL FRF matrix tells us how much response we get in mode x if we apply a force to mode y.

So if we know exacly how much force the impacting ball is applying to each mode of the bat (the modal force), we can determine exactly how much each mode will responsd by using the modal FRF matrix.

The trouble is that we don't know the modal force. We can however estimate or measure the actual physical force and it is possible to convert this force to a modal force.

If we arrange our measured mode shapes, obatined from the modal analysis, into the columns of a matrix, then we have the "modal matrix" of the system, phi. The size of the phi matrix is n rows by m columns, where n is the number of measurement positions on the bat and m is the total number of modes we are considering. Let's assume that to all intents and purposes, the force applied by the ball on the bat occurs at a single point. We can define a vector of applied forces at each measurement point. Each element in this vector will be zero except the element correstponding to the position where the impact takes place. We can now find the modal force simply by multiplying the transpose of the modal matrix by the force vector to give us a modal force vector.

So we now have the modal force vector, F, and the modal FRF matrix, H. The modal response vector, X, is then given simply by

X=H*F

The elements of the response vector X tell you exactly what you need to know ie. the degree to which each mode is excited by the impact of the ball.

I realise that what I have said here is not enough to allow you to do the calculation, but if you read this in conjunction with a text book you may find it useful.

M

 

[Rob45]

Regarding vibration in baseball bats, there's an interesting article on Lodengraf damping in a past issue of Sound and Vibration (

http://www.sandv.com)

that addresses making a metal bat sound like a "real" baseball bat - a wooden one!
(Sorry if my biases are showing here)
The article is also available on the internet if you do a search on "Lodengraf damping" but the graphics are rather poor. It is unfortunately NOT available on the S&V website.