Technology & Engineering
Laplace Transforms
Laplace transforms are mathematical tools used to simplify and solve differential equations in engineering and technology. They convert functions of time into functions of complex frequency, making it easier to analyze and manipulate dynamic systems. By transforming differential equations into algebraic equations, Laplace transforms provide a powerful method for solving a wide range of engineering problems.
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10 Key excerpts on "Laplace Transforms"
- William Bober, Andrew Stevens(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
201 Chapter 7 Laplace Transforms 7.1 Introduction Transform techniques involve applying a mathematical operation to the equations of a problem, solving in the (presumably easier) transform domain, and then taking the inverse transform to obtain an answer in the domain of interest. Laplace trans-forms are frequently used to solve ordinary and partial differential equations. The method reduces an ordinary differential equation to an algebraic equation, which can be manipulated to a form such that the inverse transform is easily obtained. In circuit theory, any circuit containing capacitors or inductors will contain at least one differential relationship, and a large circuit may contain many more. In Chapter 6, we solved circuits directly in the time domain by integrating the dif-ferential equations in MATLAB. We now describe the Laplace transform approach [1,2]; we convert the time-domain equations to the Laplace domain, solve algebra-ically, and then take the inverse transform back to the time domain to obtain the final answer. Sometimes, we do not even bother with the final step because the result in the Laplace domain is sufficient to fully solve the problem of interest. 7.2 Laplace Transform and Inverse Transform Let f ( t ) be a causal function such that it has some defined value for all t ≥ 0 and is zero otherwise, that is, f ( t ) = 0 for t < 0. The unilateral Laplace transform is defined as L ∫ = = -∞ ( ( )) ( ) ( ) 0 f t F s e f t dt st (7.1) 202 ◾ Numerical and Analytical Methods with MATLAB F ( s ) is called the Laplace transform of f ( t ), and the symbol L is used to indi-cate the transform operation. By convention, we use lowercase letters for time domain functions and uppercase for their Laplace domain (or “ s -domain”) counterparts.
eBook - ePub- Huw Fox, William Bolton(Authors)
- 2002(Publication Date)
- Butterworth-Heinemann(Publisher)
6Laplace transform
SummaryIn order to consider the response of engineering systenns, e.g. electrical or control systems, to inputs such as step, or perhaps an impulse, we need to be able to solve the differential equation for that system with that particular form of input. As the previous chapter indicates, this can be rather laborious. A simpler method of tackling the solution is to transform a differential equation into a simple algebraic equation which we can easily solve. This is achieved by the use of the Laplace transform, the subject of this chapter.Objectives By the end of this chapter, the reader should be able to:• understand what using the Laplace transform involves; • use Laplace transform tables to convert first- and second-order differential equations into algebraic equations; • use Laplace transform tables, and where appropriate partial fractions, to convert Laplace transform equations into real world equations; • determine the outputs of systems to standard input signals such as step, impulse and ramp.6.1 The Laplace transform
In this chapter a method of solving such differential equations is introduced which transforms a differential equation into an algebraic equation. This is termed the Laplace transform . It is widely used in engineering, in particular in control engineering and in electrical circuit analysis where it is commonplace not even to write differential equations to describe conditions but to write directly in terms of the Laplace transform.We can think of the Laplace transform as being rather like a function machine (Figure 6.1 ). As input to the machine we have some function of time f (t ) and as output a function we represent as F (s ). The input is referred to as being the time domain while the output is said to be in the s-domain . Thus we take information about a system in the time domain and use our ‘machine’ to transform it into information in the s -domain. Differential equations which describe the behaviour of a system in the time domain are converted into algebraic equations in the s -domain, so considerably simplifying their solution. We can thus transform a differential equation into an s -domain equation, solve the equation and then use the ‘machine’ in inverse operation to transform the s -domain equation back into a time-domain solution (Figure 6.2
- Khalid Khan, Tony Lee Graham(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
8 Laplace TransformsThe Laplace Transform is an integral transform named after its founder Pierre-Simon Laplace. It takes a function of continuous time t (t > 0) to a function of a complex variable s (frequency). As discussed earlier in Chapter 7 , when modeling real world problems, the formulation of differential equations naturally arises in many different fields of engineering.
The Laplace transform is a very important tool in engineering disciplines as it enables the following:8.1 Why Do We Need the Laplace Transform?- It helps to solve linear differential equations with given initial conditions, for systems that can be described by the following types of equations:
a8.1+ bd 2ydx 2+ c y = Ed yd x8.1
- The method is also particularly useful if the inputs to the differential equations, that is, the E term in Equation 8.1 are discontinuous inputs like the unit step function.
- Incorporates the initial conditions at the start of the solution to the problem.
- In systems engineering, the system is broken down into components as blocks. Each block can be represented in the s -domain and then manipulated.
8.2 Derivation from a Power SeriesMost mathematics textbooks will start with the formula for the Laplace transform without any reference to how it comes about mathematically. Here, it is more appropriate to first consider how the Laplace transform is derived.Starting with a discrete power series , this can be written as follows:∑ 0 ∞a nx n=a 0+a 1x +a2x 2+ …And for some a n this series can be written in closed form as, say, A (x ):∑ 0 ∞a nx n= A ( x )Using a slightly different notation for the coefficients a n = a (n ) this becomes8.2∑ 0 ∞a ( n )x n= A ( x )8.2So, different values of a (n ) can produce a different closed form sum A (x ). Some examples of using specific a (n
Available until 9 Apr |Learn more- Carlos A. Smith, Scott W. Campbell(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
111 4 Laplace Transforms The Laplace transform is a powerful tool for solving linear differential equations with con-stant coefficients and to handle several of them simultaneously. This last property devel-ops from the fact that in solving differential equations, the Laplace transform method first converts them into algebraic equations, and the resulting equations are then manipulated algebraically before obtaining the final result; algebraic manipulations are much easier, and more often possible, than dealing directly with differential equations. It is obvious at this stage that the reader may be thinking that the word Laplace denotes the name of the individual that developed the method, and this is indeed the case. But, why the term transform ? The reason for this term was hinted at in the above paragraph when we wrote “the Laplace transform method first converts them into algebraic equa-tions”; the word converts could be changed to transforms . The best way to explain the mean-ing of a transformation is by recalling the development and use of logarithms. Logarithms were independently developed by John Napier and Joost Burgi in the early seventeenth century to simplify many mathematical calculations in astronomy and navigation; Napier later collaborated with Henry Briggs, who developed most of the logarithm tables after Napier’s death. For example, consider the multiplication of 43,567 times 99,876, with the result of 4,351,297,692. This is an easy enough calculation with calculators, but back in the seven-teenth century there were no calculators, computers, or even slide rules. (By the way, slide rules were developed after the logarithms were developed—they are based on the loga-rithm scale.) With the use of logarithms, this calculation is easily obtained by the following: X = log(43,567) + log(99,876) X = 4.639157… + 4.999461… = 9.63861… Answer = antilog( X ) = antilog(9.63861…) = 4,351,297,692 which is the correct value.
- Javier Kypuros(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Chapter 5 Laplace Transforms Knowing now how to derive models in the form of systems of linear differ-ential equations, we transition our attention to the analysis and solution of such models. Though the equations are differential, the Laplace transform provides a means of converting equations in the time domain to algebraic equations in the s -domain where the models become algebraic. Consider the following questions: B What are some commonly recurring functions in dynamic systems and their Laplace Transforms? B How can Laplace Transforms be used to solve for a dynamic response in time domain? B What purpose does the Laplace transform play in analyzing the time domain response? B What are the relationships between the time domain and s -domain? 5.1 Introduction In the previous chapter we synthesized bond graphs to systematically de-rive the differential equations. The resulting state-space models can be used to simulate dynamic responses with the aid of MATLAB. With the aid of Laplace Transforms and, as we will see later, Linear Algebra, the resulting differential equations can be used to analyze and solve for system responses. Laplace Transforms are one of the most important tools for analyzing and cal-culating responses of linear systems. The fundamental advantage of using Laplace Transforms is that they refashion a differential equation problem into 151 152 CHAPTER 5. Laplace Transforms an algebraic problem, enabling us to take advantage of the many tools from Algebra that we are already familiar with. In this chapter we will review the Laplace transform. In particular, we will examine the transforms of commonly recurring mathematical functions used in the modeling of physical dynamic systems. As will be discussed, several Laplace transform theorems exist which enable one to solve differ-ential equations and assess initial and final conditions in the time domain.
eBook - PDFSignals and Systems
A Primer with MATLAB
- Matthew N. O. Sadiku, Warsame Hassan Ali(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
106 Signals and Systems: A Primer with MATLAB® 3.1 INTRODUCTION We saw in Chapter 2 that system analysis in the time-domain involves the evalu-ation of the convolution integral or sum since the response of the system is given by the convolution of the input and the impulse response This chapter introduces the Laplace transform , a very powerful, alternative tool for analyzing systems The Laplace transform is named after Pierre Simon Laplace (1749–1827), the French astronomer and mathematician In contrast to the time-domain models studied in the previous chapter, the Laplace transform is a frequency-domain representation that makes analysis and design of linear systems simpler The Laplace transform is a well-established tool in analyzing continuous-time, linear systems It is important for a number of reasons First, it is applicable to a wider variety of inputs The Laplace transform of an unbounded signal can be found Second, Laplace transform is powerful for providing us in one single operation the complete response, that is, the steady-state plus transient or homogeneous, and par-ticular Third, it automatically includes the initial conditions in the system analysis It allows us to convert ordinary differential equations into algebraic equations, which are easier to manipulate and solve It converts convolution into a simple multiplica-tion Fourth, we can apply Laplace transform to generate the transfer function repre-sentation of a continuous-time LTI system The chapter begins with the definition of the Laplace transform and uses that defi-nition to derive the transform of some basic, important functions We will consider some properties of Laplace transform which are helpful in obtaining the Laplace transform of other functions We then consider the inverse Laplace transform We apply all these to solving integro-differential equations and system analysis, espe-cially
- Mangey Ram, J. Paulo Davim, Mangey Ram, J. Paulo Davim(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
chapter oneApplication of the Laplace transform in problems of studying the dynamic properties of a material system and in engineering technologies
Lubov Mironova Russian University of Transport (MIIT) Leonid Kondratenko Moscow Aviation Institute (State National Research University) Contents 1.1 Designation 1.2 Laplace transform and operations mapping 1.3 Linear substitutions 1.4 Differentiation and integration 1.5 Multiplication and curtailing 1.6 The image of a unit function and some other simple functions 1.7 Examples of solving some problems of mechanics 1.8 Laplace transform in problems of studying oscillation of rods 1.9 Relationship between the velocities of the particles of an elementary volume of a cylindrical rod with stresses 1.10 An inertial disk rotating at the end of the rod 1.11 Equations of torsional oscillations of a disk 1.12 Equations of longitudinal oscillations of a disk 1.13 Application of the Laplace transform in engineering technology 1.13.1 Method of studying oscillations of the velocities of motion and stresses in mechanisms containing rod systems 1.13.2 Features of functioning of a drive with a long force line 1.13.3 Investigation of dynamic features of the system in the technologies of deephole machining ReferencesThis chapter is written by engineers for engineers. The authors try to convey to the reader the simplicity and accessibility of the methods in a concise form with the illustration of the calculation schemes. For a more extensive study of the stated problems of mathematical modeling, at the end of the chapter are given the literature sources, from which the reader can obtain the necessary additional explanations. The list of authors includes well-known scientists in the field of mathematics and mechanics – G. Doetsch, A.I. Lur’e, L.I. Sedov, V.A. Ivanov, and B.K. Chemodanov. In compiling the theoretical material, we refer to the authors mentioned. This chapter reflects the experience of lecturing on mathematical methods of modeling, as well as the personal participation of the authors in the work in this technical field.
- Robert B. Northrop(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
3 -1 3 The Laplace Transform and Its Applications 3.1 Introduction The Laplace transform was developed in the early nineteenth century by Pierre Simon Laplace (1749–1827), a gifted French mathematician, astronomer, cosmologist, and phys-icist. Laplace studied many natural phenomena, and was gifted to be able to invent novel mathematical tools to describe and explain his results. He is best known today for devis-ing the concept of spherical harmonics, the concept of potential in three-dimensional (3-D) space, V ( x , y , z ), and the “Laplacian operator” [Equation (3.1)] = ∂ ∂ ∂ ∂ ∂ ∂ = 2 2 2 0 V V x V y V z 2 2 2 2 (3.1) and, of course, the transform that bears his name. The Laplace transform allows continuous time (or space) functions to be written as functions in the (vector) frequency domain, that is, f ( t ) ⇔ F ( s ), t ≥ 0. It will be seen that one advantage of frequency-domain representation by the Laplace transform is that the Laplace-transformed (LT’d) output, Y ( s ), of a linear, time-invariant (LTI) system with an LT’d impulse response, H ( s ), given an LT’d transient input, X ( s ), is simply the com-plex product, Y ( s ) = X ( s ) H ( s ). (Real convolution is not necessary.) H ( s ) is also called the LTI system’s transfer function . Y ( s ) can be inverse-transformed to give the LTI system’s output, y ( t ), t ≥ 0. Laplace transform tables are generally used. The one-sided Laplace transform is defined by the real, definite integral: L f t F s f t t st t { ( ) ( )} ( ) e d = ≡ -= -∞ ∫ 0 (3.2) where s is a complex variable (2-D vector), s ≡ σ + j ω ; σ is Re{ s }, and ω is Im{ s }. The lower limit of 0 − allows the transforming of f ( t )’s that are discontinuous at t = 0, such as the 3 -2 Signals and Systems Analysis in Biomedical Engineering exponential f ( t ) = e − at U ( t ), or an f ( t ) with an impulse at the origin.
eBook - PDF- William Bober, Chi-Tay Tsai, Oren Masory(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
249 12 Chapter Laplace Transforms 12.1 Laplace Transform and Inverse Transform Laplace Transforms [1,2] can be used to solve ordinary and partial differential equa-tions (PDEs). The method reduces an ordinary differential equation to an algebraic equation that can be manipulated to a form such that the inverse transform can be obtained from tables. The inverse transform is the solution to the differential equation. The inverse transform can also be obtained by residue theory in complex variables. The method is applicable to problems where the independent variable domain is from (0 to ∞ ). The method is particularly useful for linear, nonhomoge-neous differential equations, such as vibration problems where the forcing function is piecewise continuous. Let f ( t ) be a function defined for all t ≥ 0 ; then L ( f ( t )) = = -∞ ∫ F s e f t dt s t ( ) ( ) 0 (12.1) F ( s ) is called the Laplace Transform of f ( t ). The inverse transform of F ( s ) is defined to be the function f ( t ) , that is, L -1 ( F ( s )) = f ( t ) (12.2) We can create a table that contains both f ( t ) and the corresponding F ( s ). 250 ◾ Numerical and Analytical Methods with MATLAB Example 12.1 Let f ( t ) = 1; t ≥ 0 . Then L (1) = e dt s t -∞ ∫ 0 = – e s s s t -∞ = 0 1 (12.3) Example 12.2 Let f ( t ) = e at .
eBook - PDF- Daniel Fleisch(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
4 Applications of the Laplace Transform The previous three chapters were designed to help you understand the meaning and the method of the Laplace transform and its relation to the Fourier transform (Chapter 1), to show the Laplace transform of a few basic functions (Chapter 2), and to demonstrate some of the properties that make the Laplace transform useful (Chapter 3). In this chapter, you will see how to use the Laplace transform to solve problems in five different topics in physics and engineering. Those problems involve differential equations, so the first section of this chapter (Section 4.1) provides an introduction to the application of the Laplace transform to ordinary and partial differential equations. Once you have an understanding of the general concept of solving a differential equation by applying an integral transform, you can work through specific applications including mechanical oscillations (Section 4.2), electrical circuits (Section 4.3), heat flow (Section 4.4), waves (Section 4.5), and transmission lines (Section 4.6). Each of these applications has been chosen to illustrate a different aspect of using the Laplace transform to solve differential equations, so you may find them useful even if you have little interest in the specific subject matter. And as in every chapter, the final section (Section 4.7) of this chapter has a set of problems you can use to check your understanding of the concepts and mathematical techniques presented in this chapter. 4.1 Differential Equations It’s a common saying in physics and engineering that many of the important laws of nature are expressed as differential equations. That is true because natural laws often describe how quantities change over space and time, and the mathematical expressions for those spatial and temporal changes are ordinary 121
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