Laplace Shifting Theorem

5 Key excerpts on "Laplace Shifting Theorem"

• 

  eBook - PDF

  Advanced Engineering Mathematics, SI Edition

  • Peter O'Neil(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  As such, Laplace transforms are a useful tool for initial modeling of control loops and under-standing how physical parameters affect the response time and amplitude of a controlled process. Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 90 CHAPTER 3 The Laplace Transform EXAMPLE 3.7 To compute the transform of a shifted Heaviside function H ( t − a ) , write H ( t − a ) = H ( t − a ) f ( t − a ) , where f ( t ) = 1 for all t . By the second shifting theorem, L [ H ( t − a ) ] ( s ) = L [ H ( t − a ) f ( t − a ) ] ( s ) = e − as F ( s ) = 1 s e − as because the Laplace transform of 1 is 1 / s . EXAMPLE 3.8 Determine the Laplace transform of g ( t ) = 0 for t < 2, t 2 + 1 for t ≥ 2. To apply the second shifting theorem, write g ( t ) as a function, or perhaps sum of functions, of the form H ( t − 2 ) f ( t − 2 ) . To do this, first write t 2 + 1 as a function of t − 2: t 2 + 1 = ( t − 2 + 2 ) 2 + 1 = ( t − 2 ) 2 + 4 ( t − 2 ) + 5 . Then g ( t ) = H ( t − 2 )( t 2 + 1 ) = H ( t − 2 )( t − 2 ) 2 + 4 H ( t − 2 )( t − 2 ) + 5 H ( t − 2 ). Now apply the second shifting theorem to each term to obtain L [ g ] = L [ H ( t − 2 )( t − 2 ) 2 ] + 4 L [ H ( t − 2 )( t − 2 ) ] + 5 L [ H ( t − 2 ) ] = e − 2 s L [ t 2 ] + 4 e − 2 s L [ t ] + 5 e − 2 s L [1] = e − 2 s 2 s 3 + 4 s 2 + 5 s . As usual, any formula for a transform can be written as a formula for an inverse transform. The inverse version of the second shifting theorem is L − 1 [ e − as F ( s ) ] ( t ) = H ( t − a ) f ( t − a ). (3.7) EXAMPLE 3.9 Suppose we want L − 1 s s 2 + 4 e − 3 s . The presence of the factor e − 3 s is the tipoff that equation (3.7) may apply. From the table, read that L − 1 s s 2 + 4 = cos ( 2 t ). Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Mathematics for Engineering, Technology and Computing Science

  The Commonwealth and International Library: Electrical Engineering Division

  • Hedley G. Martin, N. Hiller(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 6 The Laplace Transformation 6.1. INTRODUCTION A mathematical transformation is a special way of changing a function, the purpose being to produce a different function, the transform, which is more useful than the original for achieving certain results. The transformation due to Laplace is the best-known integral transform in applied mathematics and it is used mainly for the solution of differential equations. An integral transform of a function f(t) has the form f f(t) K( s, t) dt, a where K is a specified function of s and t, called the kernel of the transform. Between given values of the limits a and b, the integral is determined as a function of s. It is probably true that upon first meeting the Laplace transform method for solving differential equations, most students find it less attractive than the conventional methods, such as the D operator approach, and may tend to conclude that it is inferior. This is an unfortunate attitude that should fade with experience, for the transform method is of special use when conventional methods are difficult to apply and it has valuable features in connection with many of the equations which arise in practice. The systems to which the Laplace transform technique is 213 214 MATHEMATICS FOR ENGINEERING AND TECHNOLOGY applied are usually described in terms of time as the indepen-dent variable. For this reason the functions considered in the theory and in examples are taken to be dependent upon t rather than on the more conventional x. 6.2. DEFINITION OF THE TRANSFORMATION Let f(t) be a function of t defined for all positive values of t. Multiply f(t) by the kernel e 8 `, where s is independent of t, and integrate the product with respect to t from zero to infinity. The result is a function of s, denoted by f(s) to indicate its relation to the original function f(t). This operation is the Laplace transformation of f(t) and the resulting transform f(s), or f, is also denoted by eC> { f (t)} or simply by oC(f ).

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mathematics for Engineers and Technologists

  • Huw Fox, William Bolton(Authors)
  • 2002(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  6

  Laplace transform

  Summary

  In 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

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Automotive Handbook

  • (Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Mathematics 219 Laplace transform There are many control loops in a vehicle (see Control engineering); in the engine (e. g. knock control, λ control), in the A/C unit or in the undercarriage (e. g. yaw-rate control). These control loops are often represented by linear differential equa- tions. One way to solve this differential equation is to use the exponential ap- proach (see Differential equations). Alternatively, the Laplace transform can also be utilized if initial values like y(0), y'(0) etc. are given in addition to the dif- ferential equation ([1], [2], [3]). The advan- tage to this approach is that the Laplace transformation converts the differential equation into an algebraic equation. This can usually be resolved easily according to the Laplace transform function Y(s). Then, the Laplace inverse transform must be carried out in order to find the unknown function y(x). The Laplace transform L {y(x)} or image function Y(s) of a function y(x) can be found as follows: Y(s) = L { y(x)} = ∫ 0 ∞ e −sx y(x) dx, s P ℝ. In order to solve for the integral, the abso- lute value of the function must be expo- nentially bound. s merely acts as a variable in the transformation [3]. Properties Laplace transforms of the derivatives y'(x), y''(x) are very crucial when solving differential equations. L { y'(x)} = s Y(s) − y(0) L { y''(x)} = s 2 Y(s) − s y(0) − y'(0) L { c 1 y 1 (x) + c 2 y 2 (x)} = c 1 L { y 1 (x)}+ c 2 L {y 2 (x)} (linearity) L { y (ax)} = 1 __ a Y ( s __ a ), a > 0 (Similarity theorem) L { e −ax y (x)} = Y(s + a), a > 0 (Complex shifting theorem) As you can see, the derivatives are con- verted into terms with the function Y(s). A few selected examples of the Laplace transforms of functions have been com- piled in Table 14.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Advanced Mathematical Techniques in Engineering Sciences

  • Mangey Ram, J. Paulo Davim, Mangey Ram, J. Paulo Davim(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  chapter one

  Application 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 References

  This 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.

  Sign up to read

  Learn more about book