7.1 INTRODUCTION

Integral transforms are considered to be operational methods or operational calculus methods that are developed for the efficient solution of differential and integral equations. In these methods, the operations of differentiation and integration are symbolized by algebraic operators. Oliver Heaviside (1850–1925) was the first person to develop and use the operational methods for the solution of the telegraph equation and the second‐order hyperbolic partial differential equations with constant coefficients in 1892 [1]. However, his operational methods were based mostly on intuition and lacked mathematical rigor. Although subsequently, the operational methods have developed into one of most useful mathematical methods, contemporary mathematicians hardly recognized Heaviside's work on operational methods, due to its lack of mathematical rigor.

Subsequently, many mathematicians have tried to interpret and justify Heaviside's work. For example, Bromwich and Wagner tried to justify Heaviside's work on the basis of contour integration [2,3]. Carson attempted to derive the operational method using an infinite integral of the Laplace type [4]. Van der Pol and other mathematicians tried to derive the operational method by employing complex variable theory [5]. All these attempts proved successful in establishing the mathematical validity of the operational method in the early part of the twentieth century. As such, the modern concept of the operational method has a rigorous mathematical foundation and is based on the functional transformation provided by Laplace and Fourier integrals.

In general, if a function , defined in terms of the independent variable t, is governed by a differential equation with certain initial or boundary conditions, the integral transforms convert into defined by

(7.1)

where s is a parameter, is called the kernel of the transformation, and and are the limits of integration. The transform is said to be finite if and are finite. Equation (7.1) is called the integral transformation of . It converts a differential equation into an algebraic equation in terms of the new, transformed function . The initial or boundary conditions will be accounted for automatically in the process of conversion to an algebraic equation. The resulting algebraic equation can be solved for without much difficulty. Once is known, the original function can be found by using the inverse integral transformation.

If a function f, defined in terms of two independent variables, is governed by a partial differential equation, the integral transformation reduces the number of independent variables by one. Thus, instead of a partial differential equation, we need to solve only an ordinary differential equation, which is much simpler. A major task when using the integral transform method involves carrying out the inverse transformation. The transform and its inverse are called the transform pair. The most commonly used integral transforms are the Fourier and Laplace transforms. The application of both these transforms for the solution of vibration problems is considered in this chapter.

7.2 FOURIER TRANSFORMS

7.2.1 Fourier Series

In Section 1.10 we saw that the Fourier series expansion of a function that is periodic with period and contains only a finite number of discontinuities is given by

(7.2)

where the coefficients and are given by

(7.3)

Using the identities

(7.4)

Eq. (7.2) can be expressed as

(7.5)

where . By defining the complex Fourier coefficients and as

(7.6)

Eq. (7.5) can be expressed as

(7.7)

where the Fourier coefficients can be determined using Eqs. (7.3) as

(7.8)

Using Eq. (7.8), Eq. (7.7) can be written as

(7.9)

7.2.2 Fourier Transforms

When the period of the periodic function in Eq. (7.9) is extended to infinity, the expansion will be applicable to nonperiodic functions as well. For this, let and . As and the subscript n need not be used since the discrete value of becomes continuous. By using the relations and as , Eq. (7.9) becomes

(7.10)

Equation (7.10), called the Fourier integral, is often expressed in the form of the following Fourier transform pair:

(7.11)

(7.12)

where is called the Fourier transform of and is called the inverse Fourier transform of . In Eq. (7.12), can be considered as the harmonic contribution of the function in the frequency range to . This also denotes the limiting value of as , as indicated by Eq. (7.8). Thus, Eq. (7.12) denotes an infinite sum of harmonic oscillations in which all frequencies from to are represented.

Notes

1. By rewriting Eq. (7.10) as
   (7.13)

   the Fourier transform pair can be defined in a symmetric form as

   (7.14)
   (7.15)

   It is also possible to define the Fourier transform pair as

   (7.16)
   (7.17)
2. The Fourier transform pair corresponding to an even function can be defined as follows:
   (7.18)
   (7.19)

   The Fourier sine transform pair corresponding to an odd function can be defined as

   (7.20)
   (7.21)
3. The Fourier transform pair is applicable only to functions that satisfy Dirichlet's conditions in the range . A function is said to satisfy Dirichlet's conditions in the interval if (a) has only a finite number of maxima and minima in and has only a finite number of finite discontinuities with no infinite discontinuity in . As an example, the function satisfies Dirichlet's conditions in the interval , whereas the function does not satisfy Dirichlet's conditions in any interval containing the point because has an infinite discontinuity at .

7.2.3 Fourier Transform of Derivatives of Functions

Let the Fourier transform of the jth derivative of the function be denoted as . Then, by using the definition of Eq. (7.11),

(7.22)

Assuming that the derivative of is zero as , Eq. (7.22) reduces to

(7.23)

Again assuming that all derivatives of order are zero as , Eq. (7.23) yields

(7.24)

where is the complex Fourier transform of given by Eq. (7.11).

7.2.4 Finite Sine and Cosine Fourier Transforms

The Fourier series expansion of a function in the interval is given by [using Eq. (1.32)]

(7.25)

where

(7.26)

Using Eqs. (7.25) and (7.26), the finite cosine Fourier transform pair is defined as

(7.27)

(7.28)

A similar procedure can be used to define the finite sine Fourier transforms. Starting with the Fourier sine series expansion of a function defined in the interval (using Eq. [(7.32)]), we obtain

(7.29)

where

(7.30)

the finite sine Fourier transform pair is defined as

(7.31)

(7.32)

When the independent variable t is defined in the range instead of , the finite cosine transform is defined as

(7.33)

where is yet unspecified. Defining a new variable y as so that , Eq. (7.33) can be rewritten as

(7.34)

where

(7.35)

If or , then

(7.36)

Returning to the original variable t, we define the finite cosine transform pair as

(7.37)

(7.38)

Similarly, the finite sine Fourier transform pair is defined as

(7.39)

(7.40)

7.3 FREE VIBRATION OF A FINITE STRING

Consider a string of length l under tension P and fixed at the two endpoints and . The equation of motion governing the transverse vibration of the string is given by

(7.41)

By redefining the spatial coordinate x in terms of p as

(7.42)

Eq. (7.41) can be rewritten as

(7.43)

We now take finite sine transform of Eq. (7.43). According to Eq. (7.31), we multiply Eq. (7.43) by sin np and integrate with respect to p from 0 to :

(7.44)

where

(7.45)

Since the string is fixed at and , the first term on the right‐hand side of Eq. (7.45) vanishes, so that

(7.46)

Thus, Eq. (7.44) becomes

(7.47)

Defining the finite Fourier sine transform of as (see Eq. [7.31])

(7.48)

Eq. (7.47) can be expressed as an ordinary differential equation as

(7.49)

The solution of Eq. (7.49) is given by

or

(7.50)

where the constants and or and can be determined from the known initial conditions of the string.

Let the initial conditions of the string be given by

(7.51)

(7.52)

In terms of the finite Fourier sine transform defined by Eq. (7.48), Eqs. (7.51) and (7.52) can be expressed as

(7.53)

(7.54)

where

(7.55)

or

(7.56)

(7.57)

or

(7.58)

Equations (7.53), (7.54), and (7.50) lead to

(7.59)

(7.60)

Thus, the solution, Eq. (7.50), becomes

(7.61)

The inverse finite Fourier sine transform of is given by (see Eq. [7.32])

(7.62)

Substituting Eq. (7.61) into (7.62), we obtain

(7.63)

Using Eqs. (7.56) and (7.58), Eq. (7.63) can be expressed in terms of x and t as

(7.64)

7.4 FORCED VIBRATION OF A FINITE STRING

Consider a string of length l under tension P, fixed at the two endpoints and , and subjected to a distributed transverse force . The equation of motion of the string is given by (see Eq. [8.7])

(7.65)

or

(7.66)

where

(7.67)

As in Section (7.3), we change the spatial variable x to p as

(7.68)

so that Eq. (7.66) can be written as

(7.69)

By proceeding as in the case of free vibration (Section (7.3)), Eq. (7.69) can be expressed as an ordinary differential equation:

(7.70)

where

(7.71)

(7.72)

Assuming the initial conditions of the string to be zero, the steady‐state solution of Eq. (7.70) can be expressed as

(7.73)

The inverse finite Fourier sine transform of is given by (see Eq. [7.32])

(7.74)

or

(7.75)

7.5 FREE VIBRATION OF A BEAM

Consider a uniform beam of length l simply supported at and . The equation of motion governing the transverse vibration of the beam is given by (see Eq. [3.19])

(7.76)

where

(7.77)

The boundary conditions can be expressed as

(7.78)

(7.79)

We take the finite Fourier sine transform of Eq. (7.76). For this, we multiply Eq. (7.76) by and integrate with respect to x from 0 to l:

(7.80)

Here

(7.81)

In view of the boundary conditions of Eq. (7.79), Eq. (7.81) reduces to

(7.82)

Again, using integration by parts, the integral on the right‐hand side of Eq. (7.82) can be expressed as

(7.83)

in view of the boundary conditions of Eq. (7.78). Thus, Eq. (7.80) can be expressed as

(7.84)

Defining the finite Fourier sine transform of as (see Eq. [7.39])

(7.85)

Eq (7.84) reduces to the ordinary differential equation

(7.86)

The solution of Eq. (7.86) can be expressed as

(7.87)

or

(7.88)

Assuming the initial conditions of the beam as

(7.89)

(7.90)

the finite Fourier sine transforms of Eqs. (7.89) and (7.90) yield

(7.91)

(7.92)

where

(7.93)

(7.94)

Using the initial conditions of Eqs. (7.93) and (7.94), Eq. (7.88) can be expressed as

(7.95)

Finally, the transverse displacement of the beam, , can be determined by using the finite inverse Fourier sine transform of Eq. (7.95) as

(7.96)

which can be rewritten, using Eqs. (7.93) and (7.94), as

(7.97)

7.6 LAPLACE TRANSFORMS

The Laplace transform technique is an operational method that can be used conveniently to solve linear ordinary differential equations with constant coefficients. The method can also be used for the solution of linear partial differential equations that govern the response of continuous systems. Its advantage lies in the fact that differentiation of the time function corresponds to multiplication of the transform by a complex variable s. This reduces a differential equation in time t to an algebraic equation in s. Thus, the solution of the differential equation can be obtained by using either a Laplace transform table or the partial fraction expansion method. An added advantage of the Laplace transform method is that during the solution process, the initial conditions of the differential equation are taken care of automatically, so that both the homogeneous (complementary) solution and the particular solution can be obtained simultaneously.

The Laplace transformation of a time‐dependent function, , denoted as , is defined as

(7.98)

where L is an operational symbol denoting that the quantity upon which it operates is to be transformed by the Laplace integral

(7.99)

The inverse or reverse process of finding the function from the Laplace transform , known as the inverse Laplace transform, is donated as

(7.100)

Certain conditions are to be satisfied for the existence of the Laplace transform of the function . One condition is that the absolute value of must be bounded as

(7.101)

for some constants C and . This means that if the values of the constants C and can be found such that

(7.102)

then

(7.103)

Another condition is that the function must be piecewise continuous. This means that in a given interval, the function has a finite number of finite discontinuities and no infinite discontinuity.

7.6.2 Partial Fraction Method

In the Laplace transform method, sometimes we need to find the inverse transformation of the function

(7.120)

where and are polynomials in s with the degree of less than that of . Let the polynomial be of order n with roots , so that

(7.121)

First, let us consider the case in which all the n roots are distinct, so that Eq. (7.120) can be expressed as

(7.122)

where are coefficients. The points are called simple poles of .

The poles denote points at which the function becomes infinite. The coefficients in Eq. (7.122) can be found as

(7.123)

where is the derivative of with respect to s. Using the result

(7.124)

the inverse transform of Eq. (7.122) can be found as

(7.125)

Next, let us consider the case in which has a multiple root of order k, so that

(7.126)

In this case, Eq. (7.120) can be expressed as

(7.127)

Note that the coefficients can be determined as

(7.128)

while the coefficients , , can be found as in Eq. (7.125). Since

(7.129)

the inverse of Eq. (7.127) can be expressed as

(7.130)

7.6.3 Inverse Transformation

The inverse Laplace transformation, denoted as , is also defined by the complex integration formula

(7.131)

where is a suitable real constant, in Eq. (7.131), the path of the integration is a line parallel to the imaginary axis that crosses the real axis at Re and extends from to . We assume that is an analytic function of the complex variable s in the right half‐plane Re and all the poles lie to the left of the line . This condition is usually satisfied for all physical problems possessing stability since the poles to the right of the imaginary axis denote instability. The details of evaluation of Eq. (7.131) depend on the nature of the singularities of .

The path of the integration is the straight line , as shown in Fig. 7.2, in the complex s plane, with equation and Re is chosen so that all the singularities of the integrand of Eq. (7.131) lie to the left of the line . The Cauchy‐residue theorem is used to evaluate the contour integral as

(7.132)

where and the integral over tends to zero in most cases. Thus, Eq. (7.131) reduces to the form

(7.133)

Example 7.5 illustrates the procedure of contour integration.

7.7 FREE VIBRATION OF A STRING OF FINITE LENGTH

In this case, the equation of motion is

(7.134)

If the string is fixed at and , the boundary conditions are

(7.135)

(7.136)

Let the initial conditions of the string be given by

(7.137)

(7.138)

Applying Laplace transforms to Eq. (7.134), we obtain

(7.139)

where

(7.140)

Taking finite Fourier sine transform of Eq. (7.139), we obtain

(7.141)

where

(7.142)

(7.143)

(7.144)

with

(7.145)

Eq. (7.141) gives

(7.146)

Performing the inverse finite Fourier sine transform of Eq. (7.146) yields

(7.147)

Finally, by taking the inverse Laplace transform of in Eq. (7.147), we obtain

(7.148)

7.8 FREE VIBRATION OF A BEAM OF FINITE LENGTH

The equation of motion for the transverse vibration of a beam is given by

(7.149)

where

(7.150)

For free vibration, is assumed to be harmonic with frequency :

(7.151)

so that Eq. (7.149) reduces to an ordinary differential equation:

(7.152)

where

(7.153)

By taking Laplace transforms of Eq. (7.152), we obtain

(7.154)

or

(7.155)

where , and denote the deflection and its first, second, and third derivative, respectively, at . By noting that

(7.156)

(7.157)

(7.158)

(7.159)

the inverse Laplace transform of Eq. (7.155) gives

(7.160)

7.9 FORCED VIBRATION OF A BEAM OF FINITE LENGTH

The governing equation is given by

(7.161)

where denotes the time‐varying distributed force. Let the initial deflection and velocity be given by and , respectively. The Laplace transform of Eq. (7.161), with respect to t with s as the subsidiary variable, yields

(7.162)

or

(7.163)

where

(7.164)

(7.165)

Again, by taking the Laplace transform of Eq. (7.163) with respect to x with p as the subsidiary variable, we obtain

or

(7.166)

where , , , and denote the Laplace transforms with respect to t of and respectively, at . Next, we perform the inverse Laplace transform of Eq. (7.166) with respect to x. For this, we use Eqs. (7.156) –(7.159) and express the inverse transform of Eq. (7.166) as

(7.167)

Finally, we perform the inverse Laplace transform of Eq. (7.167) with respect to t to find the desired solution, . The procedure is illustrated in Example 7.7.

7.10 RECENT CONTRIBUTIONS

REFERENCES

1. 1 O. Heaviside, Electromagnetic Theory, 1899; reprint by Dover, New York, 1950.
2. 2 T. Bromwich, Normal coordinates in dynamical systems, Proceedings of the London Mathematical Society, Ser. 2, Vol. 15, pp. 401–448, 1916.
3. 3 K.W. Wagner, Über eine Formel von Heaviside zur Berechnung von Einschaltvorgangen, Archiv für Elektrotechnik, Vol. 4, pp. 159–193, 1916.
4. 4 J.R. Carson, On a general expansion theorem for the transient oscillations of a connected system, Physics Review, Ser. 2, Vol. 10, pp. 217–225, 1917.
5. 5 B. Van der Pol, A simple proof and extension of Heaviside's operational calculus for invariable systems, Philosophical Magazine, Ser. 7, pp. 1153–1162, 1929.
6. 6 L. Debnath, Integral Transforms and Their Applications, CRC Press, Boca Raton, FL, 1995.
7. 7 I.N. Sneddon, Fourier Transforms, McGraw‐Hill, New York, 1951.
8. 8 M.R. Spiegel, Theory and Problems of Laplace Transforms, Schaum, New York, 1965.
9. 9 W.T. Thomson, Laplace Transformation, 2nd ed., Prentice‐Hall, Englewood Cliffs, NJ, 1960.
10. 10 C.J. Tranter, Integral Transforms in Mathematical Physics, Methuen, London, 1959.
11. 11 E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, New York, 1948.
12. 12 W. Nowacki, Dynamics of Elastic Systems, translated by H. Zorski, Wiley, New York, 1963.
13. 13 J.W. Cooley, P.A.W. Lewis, and P.D. Welch, The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine and Laplace transforms, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 315–337, 1970.
14. 14 M.H. Cobble and P.C. Fang, Finite transform solution of the damped cantilever beam equation having distributed load, elastic support, and the wall edge elastically restrained against rotation, Journal of Sound and Vibration, Vol. 6, No. 2, pp. 187–198, 1967.
15. 15 G.R. Sharp, Finite transform solution of the vibrating annular membrane, Journal of Sound and Vibration, Vol. 6, No. 1, pp. 117–128, 1967.
16. 16 G.R. Sharp, Finite transform solution of the symmetrically vibrating annular membrane, Journal of Sound and Vibration, Vol. 5, No. 1, pp. 1–8, 1967.
17. 17 G.L. Anderson, On the determination of finite integral transforms for forced vibrations of circular plates, Journal of Sound and Vibration, Vol. 9, No. 1, pp. 126–144, 1969.
18. 18 G.L. Anderson, On Gegenbauer transforms and forced torsional vibrations of thin spherical shells, Journal of Sound and Vibration, Vol. 12, No. 3, pp. 265–275, 1970.
19. 19 S.C. Tsai, E.C. Ong, B.P. Tan, and P.H. Wong, Application of the z‐transform method to the solution of the wave equation, Journal of Sound and Vibration, Vol. 19, No. 1, pp. 17–20, 1971.

PROBLEMS

1. 7.1 Find the Fourier transforms of the following functions:
   1. 
   2. 
2. 7.2 Find the Fourier cosine transforms of the following functions:
   1. 
   2. 
3. 7.3 Find the Fourier sine transforms of the following functions:
   1. 
   2. 
4. 7.4 Find the Fourier cosine transforms of the following functions:
   1. 
   2. 
5. 7.5 Find the Fourier sine transforms of the following functions:
   1. 
   2. 
6. 7.6 Find the Laplace transforms of the following functions:
   1. 
   2. 
   3. 
   4. 
7. 7.7 Find the Laplace transforms of the following functions:
   1. 
   2. 
8. 7.8 Find the Laplace transforms of the following functions:
   1. 
   2. Heaviside's unit step function:
      
9. 7.9 A single‐degree‐of‐freedom spring–mass–damper system is subjected to a displacement and velocity at time . Determine the resulting motion of the mass (m) using Laplace transforms. Assume the spring and damping forces to be kx and , where k is the spring constant and is the velocity of the mass, and the values of m, c and k as 1, 20 and 500, respectively.
10. 7.10 Derive an expression for the response of a uniform beam of length l fixed at both ends when subjected to a concentrated force at . Assume the initial conditions of the beam to be zero. Use Fourier transforms.
11. 7.11 Find the response of a uniform beam of length l fixed at both the ends when subjected to an impulse at . Assume that the beam is in equilibrium before the impulse is applied. Use Fourier transforms.
12. 7.12 A semi‐infinitely long string, along the x‐axis, is initially at rest. The string is subjected to a transverse displacement of at the end for . Find the transverse displacement response of the string using Laplace transforms.
13. 7.13 Consider a finite string of length l fixed at and , subjected to tension P. Find the transverse displacement of a string that is initially at rest and subjected to an impulse at point using the Fourier transform method.
14. 7.14 Find the steady‐state transverse vibration response of a string of length l fixed at both ends, subjected to the force
    

    using the Fourier transform method.

15. 7.15 Find the Laplace transforms of the following functions:
    1. 
    2. 
    3. 
16. 7.16 Find the solution of the following differential equation using Laplace transforms:
    

    with and .

17. 7.17 The longitudinal vibration of a uniform bar of length l is governed by the equation
    

    where with E and denoting Young's modulus and the mass density of the bar respectively. The bar is fixed at and free at . Find the free vibration response of the bar subject to the initial conditions

    

    using Laplace transforms.

18. 7.18 Find the longitudinal vibration response of a uniform bar fixed at and subjected to an axial force at , using Laplace transforms. The equation of motion is given in Problem 7.17.
19. 7.19 A uniform bar fixed at , is subjected to a sudden displacement of magnitude at . Find the ensuing axial motion of the bar using Laplace transforms. The governing equation of the bar is given in Problem 7.17.
20. 7.20 A taut string of length 1, fixed at and is subjected to tension P. If the string is given an initial displacement
    

    and released with zero velocity, determine the ensuing motion of the string.