Problem 46 - Advanced Engineering Mathematics

Question

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{s e^{-\pi s / 2}}{s^{2}+4}\right\\} $$

Step-by-Step Solution

Answer

\(-u(t - \frac{\pi}{2}) \cdot \cos(2t)\)

Step 1: Identify the Type of Problem

We are given a Laplace inverse transform problem where we need to find a function of time, \( f(t) \), given its Laplace transform \( F(s) = \frac{s e^{-\pi s / 2}}{s^2+4} \). This means we need to apply the inverse Laplace transform \( \mathscr{L}^{-1} \).

Step 2: Simplify the Inverse Laplace Transform Expression

Recognize that the expression involves a shifted function in the \(s\)-domain because of the \( e^{-\pi s / 2} \) term. According to the shifting theorem (or second shifting theorem), and knowing \( e^{-as} F(s) \) corresponds to \( u(t-a) f(t-a) \) in time domain, we identify the base function \( F(s) = \frac{s}{s^2 + 4} \).

Step 3: Identify the Base Function's Inverse Transform

Now find \( \mathscr{L}^{-1}\left\{ \frac{s}{s^2 + 4} \right\} \). This is a standard Laplace form that corresponds to \( \cos(2t) \). So, \( f(t) = \cos(2t) \).

Step 4: Apply the Shifting Theorem in the Time Domain

Using the shifting theorem established in Step 2, substitute back with the shift: \( f(t) = u(t - \frac{\pi}{2}) \cdot \cos(2(t - \frac{\pi}{2})) \). The function involves the unit step \( u(t - \frac{\pi}{2}) \), which indicates \( \cos \) part starts at \( t = \frac{\pi}{2} \).

Step 5: Simplify the Final Expression

Distribute the shift in the cosine function so that it reflects the time shift: \( f(t) = u(t - \frac{\pi}{2}) \cdot \cos(2t - \pi) \). Use the identity \( \cos(\theta - \pi) = -\cos(\theta) \) to simplify: \( f(t) = -u(t - \frac{\pi}{2}) \cdot \cos(2t) \). This represents a cosine waveform starting at \( \frac{\pi}{2} \) with a negative amplitude.

Key Concepts

Laplace Transform

The Laplace Transform is a powerful mathematical tool used to transform a time-domain function into a frequency-domain representation. This method simplifies the process of analyzing complex systems, particularly in engineering and physics. It changes a function of time, denoted as \( f(t) \), into a function of a complex variable \( s \), denoted as \( F(s) \). The formula for the transformation is given by:\[F(s) = \int_0^\infty e^{-st} f(t) \, dt\]

• This transformation helps solve differential equations by converting them into algebraic equations in the \( s \)-domain.
• It is particularly beneficial for linear time-invariant systems.

When working with Laplace Transforms, we often focus on standard forms to make finding and using transformations straightforward.

Shifting Theorem

The Shifting Theorem, also known as the Second Shifting Theorem in Laplace transforms, aids in handling time-delayed systems. It relates a shift in the \( s \)-domain to a shift in the time domain. This theorem is particularly useful when there is an exponential term like \( e^{-as} \) in the Laplace-transformed function. The theorem states that:

• If \( F(s) \) is the Laplace transform of \( f(t) \), then \( e^{-as}F(s) \) is the Laplace transform of \( u(t-a)f(t-a) \).
• The unit step function \( u(t-a) \) ensures that the function \( f(t-a) \) starts at \( t = a \).

The Shifting Theorem allows us to address systems where signals are delayed, making it essential for solving practical engineering problems.

Unit Step Function

The Unit Step Function, often denoted \( u(t-a) \), is crucial when dealing with functions that start operating at a specific time. It effectively "turns on" or "activates" a function from a certain point \( t = a \). The function is defined as:- \( u(t-a) = 0 \) for \( t < a \),- \( u(t-a) = 1 \) for \( t \ge a \).

• This function is integral to applications in control systems and signal processing due to its ability to model step changes or switching actions.
• In practice, it helps control the segments of a signal that need to be processed or analyzed from a specific time onward.

Using the Unit Step Function in the time domain allows us to manage and manipulate functions that are not continuous over their entire domain.

Cosine Function

The Cosine Function is one of the fundamental trigonometric functions encountered in the analysis of periodic signals. In the context of Laplace transforms, specifically, it appears when transforming functions of the form \( \cos(bt) \). The standard Laplace transform of a cosine function is:\[\mathscr{L}\{\cos(bt)\} = \frac{s}{s^2 + b^2}\]

• This form simplifies the process of finding inverses when dealing with oscillatory signals, like waves or AC voltages.
• It's widely used in engineering fields to model periodic motions or oscillations.

When incorporating shifts or delays, as discussed with the Shifting Theorem, the cosine function becomes part of a more complex expression, highlighting its relevance in time-varying systems.