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Chapter 6: Problem 8

Gegeben ist das Funktionenpaar $$ f(t)=\sin (\omega t) \propto F(s)=\frac{\omega}{s^{2}+\omega^{2}} $$ Bestimmen Sie hieraus unter Verwendung des Ableitungssatzes für Bildfunktionen die Laplace-Transformierte von \(f_{1}(t)=t \cdot \sin (\omega t)\) und \(f_{2}(t)=t^{2} \cdot \sin (\omega t)\).

Short Answer

Expert verified
The Laplace transform of \( t \sin(\omega t) \) is \( \frac{2s\omega}{(s^{2} + \omega^{2})^2} \), and the Laplace transform of \( t^2 \sin(\omega t) \) is \( \frac{6s^{2} \omega - 2\omega^3}{(s^{2} + \omega^{2})^3} \).

Step by step solution

01

- Identify the given function

The given function is: \[ f(t) = \sin (\omega t) \propto F(s) = \frac{\omega}{s^{2} + \omega^{2}} \].
02

- Use the property of the Laplace transform of a time multiplied function

According to the property of the Laplace transform, the Laplace transform of \( t^n f(t) \) is given by: \[ (-1)^n \frac{d^n}{ds^n} \{ F(s) \}\].
03

- Apply the property for \( t \cdot \sin(\omega t) \)

Here, \( t \cdot \sin(\omega t) \) means \( n = 1 \), so we have to find the first derivative of \( F(s) \): \[ F(s) = \frac{\omega}{s^{2} + \omega^{2}} \]. Let's differentiate it: \[ \frac{d}{ds} \left( \frac{\omega}{s^{2} + \omega^{2}} \right) = \frac{-2s\omega}{(s^{2} + \omega^{2})^2} \].
04

- Determine the Laplace transform for \( t \sin(\omega t) \)

Thus, the Laplace transform of \( t \sin(\omega t) \) is: \[ \mathcal{L}\{t \sin(\omega t)\} = -\frac{d}{ds} \left( \frac{\omega}{s^{2} + \omega^{2}} \right) = \frac{2s\omega}{(s^{2} + \omega^{2})^2} \].
05

- Apply the property for \( t^2 \sin(\omega t) \)

Here, \( t^2 \sin(\omega t) \) means \( n = 2 \), so we have to find the second derivative of \( F(s) \): \[ \frac{d^2}{ds^2} \left( \frac{\omega}{s^{2} + \omega^{2}} \right) \]. First, let's find the first derivative result: \[ \frac{d}{ds} \left( \frac{\omega}{s^{2} + \omega^{2}} \right) = \frac{-2s\omega}{(s^{2} + \omega^{2})^2} \]. Now, differentiate again: \[ \frac{d}{ds} \left( \frac{-2s\omega}{(s^{2} + \omega^{2})^2} \right) = \frac{-2\omega (s^{2} + \omega^{2})^2 - 2s\omega \cdot 2(s^{2} + \omega^{2}) \cdot 2s}{(s^{2} + \omega^{2})^4} = \frac{-2\omega (s^{2} + \omega^{2}) + 8s^{2}\omega}{(s^{2} + \omega^{2})^3} = \frac{6s^2 \omega - 2\omega \omega^2}{(s^{2} + \omega^{2})^3} \].
06

- Determine the Laplace transform for \( t^2 \sin(\omega t) \)

Thus, the Laplace transform of \( t^2 \sin(\omega t) \) is: \[ \mathcal{L}\{t^2 \sin(\omega t)\} = (-1)^2 \frac{d^2}{ds^2} \left( \frac{\omega}{s^{2} + \omega^{2}} \right) = \frac{6s^2-2\omega^2 \omega}{(s^2 + \omega^{2})^3} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are equations that involve derivatives of a function. They are essential for describing various physical and engineering systems. For instance, the motion of a pendulum or the growth rate of bacteria can be modeled using differential equations.
In the exercise, we're given a differential equation related to trigonometric functions. We then use the Laplace transform to solve it. By transforming a differential equation with the Laplace transformation, we turn it into an algebraic equation, which is often easier to solve.
The solution involves finding the Laplace transform of functions like \( t \times \sin(\omega t) \) and \( t^2 \times \sin(\omega t) \). These operations are crucial in many fields, including physics, control engineering, and signal processing.
Time-domain Analysis
Time-domain analysis allows us to understand how a system behaves over time. It involves studying the signal or function as it changes. For example, observing the output of a circuit over time is a time-domain analysis.
In our given exercise, functions like \(\sin(\omega t)\) and \( t \times \sin(\omega t)\) are defined in the time domain. Using the Laplace transform, we can convert these time-domain functions into equivalents in the s-domain (or frequency domain).
This conversion helps in analyzing systems that would be difficult to analyze only in the time domain due to the complexity added by derivatives and integrals.
Frequency-domain Analysis
Frequency-domain analysis studies functions or signals in terms of their frequency components. Instead of looking at how a signal changes over time, we see how much of each frequency component is present.
The Laplace transform is a valuable tool for frequency-domain analysis as it converts functions from the time domain to the frequency domain. This can simplify the analysis and design of systems, particularly in electrical engineering.
In the exercise, by finding the Laplace transform of \( t \sin(\omega t) \) and \( t^2 \sin(\omega t) \), we transfer them from the time domain to the frequency domain, making them easier to handle for further analysis.
Engineering Mathematics
Engineering mathematics encompasses the application of mathematical techniques to problems of engineering. It includes calculus, differential equations, and Laplace transforms.
In this exercise, we apply engineering mathematics to solve specific problems involving sine functions and their products with time. The use of the Laplace transform here makes it easier to solve the differential equations that describe the system.
Understanding these concepts provides engineers with powerful tools to design and analyze systems ranging from electrical circuits to mechanical structures and even control systems in various industries.

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Most popular questions from this chapter

Lösen Sie die folgenden Anfangsweriprobleme: a) \(y^{\prime}-y=\mathrm{e}^{\prime}, \quad y(0)=1\) b) \(y^{\prime}+3 y=-\cos t, \quad y(0)=5\) c) \(y^{\prime}-5 y=2 \cdot \cos t-\sin (3 t), \quad y(0)=0\)

Lösen Sie die inhomogene lineare Differentialgleichung 2. Ordnung \(y^{\prime \prime}+2 y^{\prime}+y=\) \(t+\sin t\) für die Anfangswerte \(y(0)=1\), \(y^{\prime}(0)=0\) a) durch Aufsuchen einer partikulären Lösung, b) mit Hilfe der Laplace-Transformation.

Lösen Sie die folgenden linearen Differentialgleichungen 1. Ordnung mit konstanten Koefizienten: a) \(y^{\prime}-3 y=4 t \cdot \mathrm{e}^{\prime}\) b) \(y^{\prime}-4 y=5 \cdot \sin t\)

Lösen Sie die folgenden Schwingungsprobleme: a) \(\ddot{x}+a^{2}, x=0, \quad x(0)=0, \quad \dot{x}(0)=v_{0} \quad(a \neq 0)\) b) \(\ddot{x}+4 \dot{x}+29 x=0, \quad x(0)=1, \quad \dot{x}(0)=-2\) c) \(\ddot{x}+\dot{x}+0,25 x=0, \quad x(0)=1, \quad \dot{x}(0)=-1\) d) \(\ddot{x}+7 \dot{x}+12 x=0, \quad x(0)=5, \quad \dot{x}(0)=0\)

Bestimmen Sie mit Hilfe der Definitionsgleichung der Laplace-Transformation die Bildfunktionen der folgenden Originalfunktionen: a) \(f(t)=\cos (\omega t)\) b) \(f(t)=2 t+\mathrm{e}^{-4 t}\) c) \(f(t)=\mathrm{e}^{-\phi t} \cdot \sin (\omega t)\) d) \(f(t)=\sinh (a t)\) e) \(f(t)=t^{3}\) f) \(f(t)=\sin ^{2} t\)
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