12.11. Relationship with Unit Step Function (Heaviside Function)

The Dirac delta function is closely related to the Heaviside step function, which is defined as:

$u(x) = \begin{cases} 0, & x < 0 \\ 1, & x \geq 0 \end{cases}$

The Heaviside step function, denoted by u(x), is a piecewise function that takes the value 0 for all negative values of x and 1 for all non-negative values of x. It is used to represent situations where a quantity suddenly starts or stops at a certain point. This function is crucial in control systems and signal processing because it models the transition between two states.

Think of the Heaviside step function as a light switch. When the switch is off (x < 0), the light is off (0). When the switch is turned on (x ≥ 0), the light comes on (1). This change represents a sudden shift from one state to another, just like the Heaviside function jumps from 0 to 1.

The delta function is the derivative of the Heaviside function in the distributional sense: $\frac{du(x)}{dx} = \delta(x)$

In a mathematical sense, the Dirac delta function, denoted as δ(x), represents an instantaneous change or impulse at the origin. When we say that the delta function is the derivative of the Heaviside function, we mean that if you take the Heaviside function and find its rate of change, you get the delta function. This is significant because it implies that the delta function 'captures' the sudden change that occurs at x = 0, illustrating the concept of instantaneous loading in physical systems.

Consider a movie showing a door opening. The Heaviside function represents the door being closed (u(x) = 0) and then abruptly opening (u(x) = 1). The activity of the door opening can be likened to the Dirac delta function: it's a sudden event. If we were to plot the speed of the door's opening, we'd see a spike at the moment it opens — this spike represents the delta function, indicating a sudden change in state.

This relationship is particularly useful in modeling sudden application of loads in structural systems, such as the instant engagement of a support or switch in system behavior.

Engineers often need to model scenarios where forces or loads are applied all at once, such as a weight suddenly being added to a bridge or a machine starting up. The relationship between the Dirac delta function and Heaviside step function allows engineers to use delta functions to represent these instantaneous point forces. This helps in analyzing how structures react to sudden loads and in designing them to withstand such impacts.

Imagine a trampoline. When someone jumps onto it (the sudden load), it immediately starts to compress (the response). This instantaneous impact can be modeled using the Dirac delta function to represent the jump, while the bounce back after the jump corresponds to the behavior captured by the Heaviside function. Engineers can then analyze how much the trampoline dips and how it will respond to repetitive jumps by understanding this relationship.

--