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Singularity issue due to Zero-frequency component

Started by Elango_Selvam555 4 years ago4 replieslatest reply 4 years ago89 views

I am a Ph.D. student working on nonlinear elastic modeling of elastic materials in the Netherlands. In my formulation, I have a singularity issue at the zero frequency component in the expression of force potential function \(\mathrm{b}(k,t)\), as follows.   

$$ f(x,t)= Q\: \cos(\omega \:t)\:\delta(x)$$

where, \(f(x,t)\) is the given input force function,   

\(Q\) is the input amplitude  

\(\cos(\omega \:t)\) is the sine signature pulse  

\(\delta(x)\) is the delta pulse acting at \(x=0\)  

The fource (vector) function is related to the scalar potential as follows:  

$$ f(x,t)= \frac{\mathrm{db}(x,t)}{\mathrm{d}x}$$

I need to compute the potential function \(b(x,t)\), by transforming the above expression in frequency domain, which becomes,

$$\tilde{f}(k,t)=-\mathrm{j\:k} \:\tilde{\mathrm{b}}(k,t) $$ 

$$\tilde{\mathrm{b}}(k,t)=\frac{\mathrm{j}}{\mathrm{k}} \:\tilde{\mathrm{f}}(k,t) $$

where \(k\) is the angular spatial frequency, \(k=2\:\pi \:(0:M-1)\:dk\), \(M\) is the number of spatial points and \(dk\) is the spatial frequency interval. When I implement the expression  \(\tilde{\mathrm{b}}(k,t)\) in Matlab, I get the infinity value as I have zero frequency component in \(k\) array (the first value of \(k=0)\).

I tried to eliminate the singularity by doing the Taylors series expansion of \(\tilde{\mathrm{f}}(k,t)\), but it wasn't useful, as I have \(\mathrm{rect(k)}\) rectangular function in the expression for \(\tilde{\mathrm{f}}(k,t)\).

I simply replace the invalid infinite value to zeros in the expression for \(\tilde{\mathrm{b}}(k,t)\), which I justify as follows: At zero frequency value, the signal starts to oscillate at the constant frequency, whose function value at zero frequency can be set to zero, which I did in \(\tilde{\mathrm{b}}(k,t)\). 

Is this correct? If not, how can I fix this infinity issue?. Any ideas/advice would be highly appreciable.         

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Reply by dgshaw6October 29, 2018

Sorry to say I don't have an answer, but rather a question to consider.

From a practical point of view, can your forcing function, applied to your non-linear elastic material, have a "DC" component?  If so, then eliminating the zero frequency case should not be valid.

I don't know much about non-linear elastics, but it seems to me that a "DC" forcing function will have a non-zero effect (at other frequencies) on the material, especially since it is non-linear.  Just a guess.

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Reply by stephanebOctober 29, 2018

Hi,

if you want your equation to be displayed inline, use:

\(  e^{i\pi}+1=0 \)

\( e^{i\pi}+1=0 \)

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Reply by Elango_Selvam555October 29, 2018

Sorry, this is my first post in this forum, so made a mistake assuming it accepts Latex format. I will correct it. Thanks for the comment. 

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Reply by stephanebOctober 29, 2018

No worries!  I saw that you made the corrections, thank you.