Relation of Transmissibility vs Transfer Function

[garymondo]

I have a good understanding of a Transfer Function (H) between two points on a system, and have measured the same for Modal analysis, or system characterization. Recently I was asked if I could calculate Transmissibility (T) from a measured H (complex).

Most of my text books and queries discuss H without reference to T. I was told that T was the ration of two Autospectra but have been unable to confirm this.

Any help or direction is greatly appreciated.

garymondo

 

[IRstuff]

I think transmissibility is the same as what you're calling a transfer function.

TTFN

FAQ731-376

 

[Transient1]

I agree with IRstuff, that transmissibility and the transfer function are functionally the same thing.

 

[MikeyP]

I am assuming that your "trasfer function H" is an FRF. i.e. H_ab is the response (displacement, velocity or acceleration) at position 'a' due to a unit excitation (force) at position 'b'

The transmissibility T_ab is the response at position b divided by the response at position a (usually where there is some excitation at position a)

so if the response is X and the force is F then

H_ab = X_a/F_b and H_bb = X_b/F_b

then the transmissibility

T_ab = X_a/X_b = H_ab/H_bb

So if you know the FRF between points a and b and the FRF at the excitation point (between points b and b) then you can calculate the transmissibility between a and b by dividing the two FRFs (remembering to do the arithmetic with complex numbers)

M

--
Dr Michael F Platten

 

[garymondo]

Thanks to all for their submission. I was referring to the frf (frequency response function), and sometimes use the term transfer function (in error) to indicate the frf. I also did some investigation on the comparison of T and H (frf) with other sources. I thought I should share my findings with you:

From my good friend, Dr. Marcos Underwood:

There is no universally accepted definition for what we call transmissibility functions. There is a good discussion of what these are typically in Chapter 2 of the Shock and Vibration Handbook and also in the link:

http://books.google.com/books?id=fH...aUdL&sig=BtxCFsEShEBPkI1bselkSoJOdL4#PPT72,M1

Generally speaking, transmissibility is typically meant as the ratio of the response of the system and the input to the system. This is the way it's used in the two references that I've given you. Because of this, it's commonly taken as square root of the ratio of the response PSD to the input PSD, but it's not universally used in this way. However, as the references discuss, the term transmissibility is usually taken as the magnitude of a certain frequency response function, which is is not always the same as the ratio of two PSDs, if there is noise present in either the input or output measurements.

Frequency response function (frf) is more generally accepted as given by the ratio of the cross-spectral density (csd) between the response and input of a system and the psd of the input of the system. Note that if there is no noise in either the response or input measurements, then the magnitude of the so obtained frf, which is usually called the transmissibility, is the same as the square root of the ratio of the response PSD and the input PSD. Because of this, we have the situation we have, where both are called transmissibility functions. But in all cases, an frf or transmissibility function is only a function of frequency.

A transfer function is more general than an frf. It's defined for the entire complex-plane and just a function of frequency. It's typically the Laplace transform of the systems impulse response function, whereas the frf is the Fourier transform of the same impulse response function. However, many people will call an frf a transfer function, which is not technically correct.

Again, thanks to all for their thoughts,
garymondo