Content deleted Content added
No edit summary |
Link suggestions feature: 3 links added. |
||
| (11 intermediate revisions by 4 users not shown) | |||
Line 1:
{{Short description|Class of discontinuous functions}}
'''Singularity functions''' are a class of [[discontinuous function]]s that contain [[Mathematical singularity|singularities]], i.e., they are discontinuous at their 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 referred to as '''singularity brackets''' . The functions are defined as:
:{| class="wikitable"
Line 33:
== 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}
Line 41:
== 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–Bernoulli beam theory]]. Here, we are using the [[sign convention]] of
[[Image:Loaded beam.svg|center|650px]]
Load distribution:
:<math>w=-3\text{ N}\langle x-0 \rangle^{-1}\ +\ 6\text{ Nm}^{-1}\langle x-2\text{ m} \rangle^0\ -\ 9\text{ N}\langle x-4\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=-3\text{ N}\langle x-0\rangle^0\ +\ 6\text{ Nm}^{-1}\langle x-2\text{ m}\rangle^1\ -\ 9\text{ N}\langle x-4\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=3\text{ N}\langle x-0\rangle^1\ -\ 3\text{ Nm}^{-1}\langle x-2\text{ m}\rangle^2\ +\ 9\text{ N}\langle x-4\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> = 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\ -\ 1\text{ Nm}^{-1}\langle x-2\text{ m}\rangle^3\ +\ \frac{9}{2}\text{ N}\langle x-4\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-2\text{ m}\rangle^4\ +\ \frac{3}{2}\text{ N}\langle x-4\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> = 0 at <var>x</var> = 4 m allows us to solve for <var>c</var> = −7 Nm<sup>2</sup>
| |||