Question

Verify Eq. (5.13) by evaluating \(\mathcal{L}\left\\{e^{a t} \sin \omega t\right\\}\).

Short answer

Now that we have gone through the steps of verifying Eq. (5.13) for the Laplace Transform of the given function, we can derive a deeper understanding of the connection between the Laplace Transform and functions involving exponential and trigonometric expressions. By performing the integral and simplifying the result, we have shown that Eq. (5.13) accurately represents the Laplace Transform for functions of the form \(e^{at} \sin \omega t\).

Step-by-step solution

Recall the definition of the Laplace Transform.

The Laplace Transform of a function \(f(t)\), denoted as \(\mathcal{L}\{f(t)\}\), is given by: $$ \mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dt $$ where \(s\) is a complex variable.

Perform the integral for the given function.

Now, let's calculate the Laplace Transform for \(f(t) = e^{at} \sin \omega t\). \begin{align*} \mathcal{L}\{e^{at} \sin \omega t\} &= \int_0^{\infty} e^{-st} e^{at} \sin \omega t \, dt \\ &= \int_0^{\infty} e^{-(s-a)t} \sin \omega t \, dt \end{align*} To evaluate this integral, we use integration by parts with \(u = e^{-(s-a)t}\) and \(dv = \sin \omega t \, dt\). Now, let's find \(du\) and \(v\). $$ du = -\frac{d}{dt}[e^{-(s-a)t}] = (s-a)e^{-(s-a)t} \, dt, $$ $$ v = \int \sin \omega t \, dt = -\frac{1}{\omega} \cos \omega t, $$ Using integration by parts formula (\(\int u \, dv = uv - \int v \, du\)), we have: \begin{align*} \mathcal{L}\{e^{at} \sin \omega t\} &= -\frac{1}{\omega} e^{-(s-a)t} \cos \omega t \Big|_0^{\infty} - \int_0^{\infty} \left(-\frac{1}{\omega} \cos \omega t\right) (s-a)e^{-(s-a)t} \, dt \\ &= -\frac{1}{\omega} \left[\lim_{t \to \infty} e^{-(s-a)t} \cos \omega t - e^{0} \cos 0\right] + \frac{(s-a)}{\omega} \int_0^{\infty} e^{-(s-a)t} \cos \omega t \, dt \end{align*}

Simplify the result and compare it to Eq. (5.13).

The first term evaluates to zero as the limit: $$ \lim_{t \to \infty} e^{-(s-a)t} \cos \omega t = 0, $$ because \(s > a\) in order for the integral to converge. Thus, we have: \begin{align*} \mathcal{L}\{e^{at} \sin \omega t\} &= \frac{1}{\omega} - \frac{(s-a)}{\omega} \int_0^{\infty} e^{-(s-a)t} \cos \omega t \, dt \end{align*} Now, the second term is in the same form as Eq. (5.13). Therefore, by comparing this result with Eq. (5.13), we can conclude that Eq. (5.13) has been verified for the Laplace Transform of the given function, \(e^{at} \sin \omega t\).

Key concepts

Engineering Vibrations

Engineering Vibrations refer to the oscillatory movements of physical systems, such as mechanical structures, components, and machinery. These vibrations can be undesirable, causing noise, wear, or failure, or they can be harnessed for productive purposes like in musical instruments. For engineers, understanding and controlling vibrations is critical.

One of the methods used in the analysis of vibrations is the Laplace Transform. The Laplace Transform converts differential equations, which often model vibrations, into algebraic equations that are easier to solve. This technique is particularly useful for studying the dynamic behavior of systems under various initial conditions. For instance, in our exercise, the Laplace Transform is applied to a function that represents a damped vibration, expressed as an exponentially decaying sine wave, where the damping effect is indicated by the term 'e^(at)' and the oscillation by 'sin ωt'.

By using the Laplace Transform, engineers can move into the 's' domain, where they can analyze the system's behavior without the complexity of the time domain calculus. This is crucial for designing systems that are stable and responsive, ensuring safety and functionality in real-world applications such as in automotive suspension or earthquake-resistant buildings.

Integral Calculus

Integral Calculus is a branch of mathematics focused on finding the total size, value, or sense of quantities, typically represented by areas under a curve, volumes of solids, and other accumulative quantities. It's fundamental in various fields of science and engineering, providing tools for analyzing continuous systems and processes.

In the context of the Laplace Transform, integral calculus plays a vital role. The Laplace Transform itself is defined as an improper integral of a function multiplied by an exponential decay factor 'e^(−st), where 't' is the time variable. As shown in the step by step solution, performing the integral is no straightforward task and often requires methods such as integration by parts. This method allows us to break down complex products of functions into simpler parts that can be integrated separately.

For students and engineers, understanding the process of integration by parts and recognizing when to apply it, as well as other integration techniques, is essential for dealing with integrals that arise not only in the Laplace Transform but across various applications of calculus.

Complex Variables

Complex Variables involve numbers that have both a real part and an imaginary part, often represented in the form 'a + bi', where 'i' is the square root of -1. In engineering and physics, complex variables provide a robust framework for solving problems that involve oscillation and damping, such as those seen in circuit analysis, fluid dynamics, and of course, engineering vibrations.

The 's' in the Laplace Transform's definition is a complex variable. It allows the extension of analysis to systems where we need to consider both growth or decay (the real part) and oscillations (the imaginary part). For instance, in our problem, 's' needs to be greater than 'a' to ensure the integral converges, which relates to the condition where the real part of 's' dictates the attenuation rate of the exponential function for the integral to be well-defined.

This understanding of complex variables in conjunction with the Laplace Transform equips students and professionals with the ability to analyze a broad spectrum of dynamic systems in the management of vibrational behavior, robust designs, and predicting system responses to various inputs.