Introduction to Macaulay’s function and Singularity functions.
Macaulay’s function and singularity function are important concepts in mathematical analysis. Macaulay’s function is a mathematical tool used to define a piecewise function that combines polynomials with different degrees. On the other hand, the singularity function is a mathematical function commonly used to represent concentrated loads or forces in structural engineering. Both of these functions play crucial roles in various areas of mathematics and engineering.
What is a step function?
There is a function called the unit step function or Heaviside function that starts after a certain time when we plot t as the x-axis and the Heaviside function as the Y-axis.
We can express the relation as H(t-a) equals zero for t is bigger or equal to zero and less than a but equals one unit when t >=a. It forms a step shape with a riser equal one which is extended for values of T >a.
In the upper part of the slide, we can see a line graph which we are familiar with. When a line of positive slope m passes by (0,0) point its equation can be expressed as y function equals y=mx.
If the same line was shifted by a distance a from the y-axis and maintains the same slope m then y is expressed as y=m(x-a). But when the line is shifted by a distance a- from the y-axis and maintains a negative slope (-m), the equation can be expressed as y=-m(x-a).
This expression we will utilize later when dealing with a triangular load. But the y-axis will be w(x) and the load expression is written as w(x)=+w/b(x-a), where the base is the load base and w is the ordinate load intensity.
The line can express a uniform load starting from distance a from the left edge and extending to the end of a beam. Its equation will be W(x)=w(x-a)^0, , where w is the intensity of the uniform load. We use the expression of w(x-a)^0 to include that the load at x<a considered zero, while for values bigger than a it is equal to w.
Represent the shear function for a beam as a group of step functions.
In the next slide, if we have a beam with length L having two supports and acted upon by a concentrated load at a distance a, from the left support and the remaining distance to the right support is b.
We can estimate the reaction at the left support will be equal to (P*(b)/L. While the reaction at the right support will be equal to P*(a)/L.
To express the shear function, by using a step function, we can consider the previous beam as an assembly of three-step functions. The first case is the shear from the left reaction force and expressed as Q(x)=+P*b/L in the form of a step function with a rise equal to P*b/L that extends to the end of the beam.
To express the load P, as a shear function it also acts as a step function starting from distance a, with a rise of -P and extends a distance (L-a). Finally, the right support reaction with a value of P*(a/L) is also a step function that starts from x=l with a shear value of -Pa/L. The previous shapes can be combined to form the shear diagram as shown.
In the next slide, Quoted from Prof. Timothy Philpot’s book, an integrated learning system. Macaulay’s functions represent quantities that begin at point a.
Before point a the function has a zero value. After point a, the function has a value of f(x-a)^n, since it is shifted by a distance a. Macaulay’s function is expressed not by parentheses but with a different form, smaller than and bigger than signs or bracketed terms. The function value is 0 when x is smaller than a and (x-a)^n for x> or equal to a.
Macaulay’s function with different n values.
In the next slide, Macaulay‘s function deals with three values of n as n=0,1 and 2 values. When n=0, the function is a straight line and y value is (x-a)^0=1 when x=a or bigger than a and zero for x<a. that expresses a uniform load.
Macaulay function when n=1, the function is expressed as an inclined line, and the y value is (x-a)^1=1 when x=a or bigger than a and zero for x<a. this function is a ramp function and expresses a linear load. The last form is when n=2, where< x-a>^2 expresses a parabolic plot. the parabolic plot can be used to explain the moment values of distributed loads acting on a beam.
Introduction to singularity functions.
In the next slide, we can see new functions that are called singularity functions, where n =-1 or n=-2. These singularity functions can be used to express concentrated loads and concentrated moments.
These functions can give the value of the load intensity in case of a concentrated load P0 acting as w(x) as =P0*<x-a>^-1 as equal 0 when x does not equal a and equals P0 when x=a, in that case n=-1.
In the case of a concentrated Moment M0 acting, the load function expressed as w(x) as =M0*<x-a>^-2 as equal 0 when x does not equal a and equals M0 when x=a. the n value equals =-2 in this case.
The first graph shows the relations between W(x) and P0*<x-a>^-1 function shifted by a distance a. While the second graph shows the relations between W(x) and M0*<x-a>^-2 function shifted by a distance a.
The difference between the integration of Macaulay’s function and the integration of the singularity functions.
In the case of Macaulay functions, where n >0, after integration we add 1 to n and divide by(n+1) same as any ordinary function. In the case of singularity functions. Where n<0, after integration, we add 1 to n, and there is no division by (n+1).
The difference of the integration between Macaulay’s function and discontinuity functions.
This is an introductory link to Macaulay’s function.
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