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# Frequency of a Cantilever Beam with Interchangeable End Attachments

## Frequency of a Cantilever Beam with Interchangeable End Attachments

(OP)
Hello. I have been trying to figure out a way to do this off and on for a while now. I was wondering if anyone here could provide some insight into something I may have missed.

I have frequency test data on a beam that is clamped at a known length with an attachment on the end. The mass and length of the beam are known. The distance from the end of the beam to the center of gravity on the attachment as well as the mass of the attachment is known. The beam is then to be clamped at a slightly different length, with a different end attachment. The new length, mass of the new attachment, and offset of center of gravity are all known. I am trying to predict the frequency in this new configuration. The beam is non-uniform in all accounts. I have pulled and tested a sample of these beams, as well as a sampling of attachments on each for testing purposes, so I have a way of checking my results.

I feel like I should be able to use the described values to determine how the beam would react without the original end attachment, and then account for the specifications on the new attachment. Below is what I have tried so far, but still have around a 3% error.

w = sqrt[3*EI / meffL3]
By using the above equation (which can be found in numerous places for equating a beam to an end mass), I got to the following.

w = sqrt[3*EI / (mbeamLbeam3 + mattachment(Lbeam+Loffset)3)]
Using this, I was able to get to an "EI" for the beam-attachment system. I then used the derived "EI", the new beam length, and the new attachment specifications in the same above equation to calculate a frequency for the new configuration.

Can anyone think of a better way to do this?

### RE: Frequency of a Cantilever Beam with Interchangeable End Attachments

I can't help but think that a dimensioned diagram might save a thousand words. If the beam is non uniform along its length and you are clamping it in a different position then its EI will change.

One way of doing this would be FEA. However a real engineer will probably be able to get close enough with a hand calc.

Cheers

Greg Locock

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### RE: Frequency of a Cantilever Beam with Interchangeable End Attachments

It is a routine matter to calculate the beam response, unless you are dealing with nanometer sized objects. soi what so special about your devices?

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