Singularity function

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

$$langle x-a angle^n = egin{cases}delta'(x-a) : & n=-2\delta(x-a) : & n=-1\0 : & nge0, x

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.

$$int^n dx = egin{cases} ^{n+1}, & n<0 \ frac{^{n+1{n+1}, & n ge 0 end{cases}

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'=frac{1}{EI}int M dx:Because the slope is not zero at x=0, a constant of integration, c, is added:$$u'=frac{1}{EI}(frac{3}{2}N^2 - 1Nm^{-1}^3 + frac{9}{2}N^2 + c),Deflection::$$u=int u' 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