Singularity function: Difference between revisions

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{{Short description|Class of discontinuous functions}}

{{Unreferenced|date=February 2011}}

'''Singularity functions''' are a class of discontinues[[discontinuous functionsfunction]]s that contain [[Mathematical singularity|singularitysingularities]], i.e., they are discontinuous at itstheir singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of [[generalized functions]] and [[Distribution (mathematics)|distribution theory]].<ref>{{citation |first=A. H.|last=Zemanian|title=Distribution Theory and Transform Analysis|publisher=McGraw-Hill Book Company| year=1965}}</ref><ref>{{citation|first=R. F.|last=Hoskins|title=Generalised Functions|publisher=Halsted Press|year=1979}}</ref><ref>{{citation |first=M.J.|last=Lighthill|title=Fourier Analysis and Generalized Functions|publisher=Cambridge University Press|year=1958}}</ref> The functions are notated with brackets, as <math>\langle x-a\rangle ^n</math> where ''n'' is an [[integer]]. The "<math>\langle \rangle</math>" are often refereedreferred to as '''singularity brackets''' . The functions are defined as:

:{| class="wikitable"

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|-

| 2

| <math>\frac{(x-a)^2}{2}H2H(x-a)\,</math>

|-

| <math>\ge 0</math>

| <math>\frac{(x-a)^n}{n+1}\,HnH(x-a)\,</math>

|}

where: {{math|''δ''(''x'')}} is the [[Dirac delta function]], also called the unit impulse. The first derivative of {{math|''δ''(''x'')}} is also called the [[unit doublet]]. The function <math>H(x)</math> is the [[Heaviside step function]]: {{math|1=''H''(''x'') = 0}} for {{math|''x'' < 0}} and {{math|1=''H''(''x'') = 1}} for {{math|''x'' > 0}}. The value of {{math|''H''(0)}} will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for {{math|1=''n=0'' = 0}} since the functions contain a multiplicative factor of {{math|''x-'' − ''a''}} for {{math|''n'' > 0}}. <br/>

where:

<math>\langle x-a\rangle^1</math> is also called the [[Ramp function]].

δ(x) is the [[Dirac delta function]], also called the unit impulse. The first derivative of δ(x) is also called the [[unit doublet]]<br/>

<math>H(x)</math> is the [[Heaviside step function]]: H(x)=0 for x<0 and H(x)=1 for x>0. The value of H(0) will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for ''n=0'' since the functions contain a multiplicative factor of ''x-a'' for n>0. <br/>

<math>\langle x-a\rangle^1</math> is also called the [[Ramp function]].

== Integration ==

Integrating <math>\langle x-a \rangle^n</math> can be done in a convenient way in which the [[constant of integration]] is automatically included so the result will be {{math|0}} at {{math|1=''x'' = ''a''}}.

<math display="block">\int\langle x-a \rangle^n dx = \begin{cases}
\langle x-a \rangle^{n+1}, & n\le< 0 \\
\frac{\langle x-a \rangle^{n+1}}{n+1}, & n \ge 0
\end{cases}</math>

== Example beam calculation ==

The deflection of a simply supported beam, as shown in the diagram, with constant cross-section and elastic modulus, can be found using [[Euler-BernoulliEuler–Bernoulli beam equation|Euler-Bernoullitheory]] beam theory. Here, we are using the [[sign convention]] of downwardsdownward forces and sagging bending moments being positive.

[[Image:Loaded beam.PNGsvg|center|650px]]

Load distribution:

:<math>w=-3N3\text{ N}\langle x-0 \rangle^{-1}\ +\ 6Nm6\text{ Nm}^{-1}\langle x-2m2\text{ m} \rangle^0\ -\ 9N9\text{ N}\langle x-4m4\text{ m}\rangle^{-1}\, -\ 6\text{ Nm}^{-1}\langle x-4\text{ m} \rangle^0\ </math>

Shear force:

:<math>S=\int w\, dx</math>

:<math>S=-3N3\text{ N}\langle x-0\rangle^0\ +\ 6Nm6\text{ Nm}^{-1}\langle x-2m2\text{ m}\rangle^1\ -\ 9N9\text{ N}\langle x-4m4\text{ m}\rangle^0\ -\ 6\text{ Nm}^{-1}\langle x-4\text{ m}\rangle^1\,</math>

Bending moment:

:<math>M = -\int S\, dx</math>

:<math>M=3N3\text{ N}\langle x-0\rangle^1\ -\ 3Nm3\text{ Nm}^{-1}\langle x-2m2\text{ m}\rangle^2\ +\ 9N9\text{ N}\langle x-4m4\text{ m} \rangle^1\ +\ 3\text{ Nm}^{-1}\langle x-4\text{ m}\rangle^2\,</math>

Slope:

:<math>u'=\frac{1}{EI}\int M\, dx</math>

:Because the slope is not zero at <var>x</var>&nbsp;=&nbsp;0, a constant of integration, <var>c</var>, is added

:<math>u'=\frac{1}{EI}\left(\frac{3}{2}\text{ N}\langle x-0\rangle^2\ -\ 1Nm1\text{ Nm}^{-1}\langle x-2m2\text{ m}\rangle^3\ +\ \frac{9}{2}\text{ N}\langle x-4m4\text{ m}\rangle^2\ +\ 1\text{ Nm}^{-1}\langle x-4\text{ m}\rangle^3\ +\ c\right)\,</math>

Deflection:

:<math>u=\int u'\, dx</math>

:<math>u=\frac{1}{EI}\left(\frac{1}{2}\text{ N}\langle x-0\rangle^3\ -\ \frac{1}{4}\text{ Nm}^{-1}\langle x-2m2\text{ m}\rangle^4\ +\ \frac{3}{2}\text{ N}\langle x-4m4\text{ m}\rangle^3\ +\ \frac{1}{4}\text{ Nm}^{-1}\langle x-4\text{ m}\rangle^4\ +\ cx\right)\,</math>

The boundary condition <var>u</var>&nbsp;=&nbsp;0 at <var>x</var>&nbsp;=4m&nbsp;4&nbsp;m allows us to solve for <var>c</var>&nbsp;=-7Nm&nbsp;−7&nbsp;Nm²

==See also==

*[[Macaulay brackets]]

*[[Macaulay's method]]

==References==

{{Reflist}}

==External links==

* [https://web.archive.org/web/20100401012230/http://www.cgl.uwaterloo.ca/~tjlahey/sfunctions.pdf Singularity Functions (Tim Lahey)]

* [http://www.cgl.uwaterloo.ca/~tjlahey/sfunctions.pdf Singularity Functions (Tim Lahey)]

* [http://www.ce.berkeley.edu/~coby/CE130/sing.pdf Singularity functions (J. Lubliner, Department of Civil and Environmental Engineering)]

* [http://www.assakkaf.com/Courses/ENES220/Lectures/Lecture17.pdf Beams: Deformation by Singularity Functions (Dr. Ibrahim A. Assakkaf)]