Singularity function

Singularity functions or singularity brackets are a notation used to describe discontinuous functions.

δ'(x) is the first derivative of δ(x), also called the unit doublet.
δ(x) is the Dirac delta function, also called the unit impulse.

Integration

Integrating $^n$ can be done in a convenient way in which the constant of integration is automatically included so the result will be 0 at x=a.

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 downwards forces and sagging bending moments being positive.

Load distribution::$w=-3N^\{-1\}\; +\; 6Nm^\{-1\}^0\; -\; 9N^\{-1\},$Shear force::$S=int\; w\; dx$:$S=-3N^0\; +\; 6Nm^\{-1\}^1\; -\; 9N^0,$Bending moment::$M\; =\; -int\; S\; dx$:$M=3N^1\; -\; 3Nm^\{-1\}^2\; +\; 9N^1,$Slope::$u\text{'}=frac\{1\}\{EI\}int\; M\; dx$:Because the slope is not zero at x=0, a constant of integration, c, is added:$u\text{'}=frac\{1\}\{EI\}(frac\{3\}\{2\}N^2\; -\; 1Nm^\{-1\}^3\; +\; frac\{9\}\{2\}N^2\; +\; c),$Deflection::$u=int\; u\text{'}\; dx$:$u=frac\{1\}\{EI\}(frac\{1\}\{2\}N^3\; -\; frac\{1\}\{4\}Nm^\{-1\}^4\; +\; frac\{3\}\{2\}N^3\; +\; cx),$The boundary condition u=0 at x=4m allows us to solve for c=-7Nm²

ee also

*Macaulay brackets