fourier tranform application in euler and bernouili beam

DYNAMIC STRAIN RESPONSE OF AN INFINITE BEAM
AND INVERSE CALCULATION OF IMPACT FORCE BY
NUMERICAL LAPLACE TRANSFORM

OBJECTIVE
› To derives a force-strain relation for the BernoulliEuler beam using the Laplace transform approach.
› The strain response and the impact force are
obtained by numerical inversion of the transformed
solution using a Laplace transform algorithm

FORCE-STRAIN RELATION
› Structure is assumed to be a narrow beam of the BernoulliEuler type.
› The applied force and the beam displacement are related by

(1)
 where El and pA are the stiffness and mass perunit length of the beam, respectively.
 F(t) is the applied force,
 v is the displacement of the beam

 x is the distance from the point of impact.
 The boundaries are considered to be at infinity

› The force-strain relation becomes

o Q1 and Q2 depend only on the material and section properties of the beam.
o e(x, t) is the strain at an arbitrary position x and h is the thickness of the beam.

› The spectral relation between force and strain is

› Consider an infinite beam shown in Fig. 1.

› The beam model assumes that only bending moment M
and shear force V resultants act at the beam section.

› Application of the Laplace transform with respect to time to
Eq.(1) under zero initial conditions yields
(2)

p is the mass per unit volume,
A is the crosssectional area,
E is the Young's modulus,
I is the moment of inertia of the cross section about the neutral axis
s is the Laplace transform parameter.

› The homogeneous solution of Eq. (2) takes the following
form

where A, B, C and D are constants.
› Since the beam is infinite then only two waves are
generated.
(3)
(4)

› Imposing continuity of displacement and slope at the
joint, and writing the equations of motion of the joint itself
gives (at x=0)

(5)

› Substituting Eqs. (3) and (4) into Eq. (5) gives the
displacement and the strain for the positive x direction (x
>0) as
(6)

› When the force spectrum ff is known, then the strain spectrum (and hence
the strain history) at an arbitrary position x can be obtained.
› The transfer function for the strain is

› If a strain history is measured at an arbitrary position x, then the force
history causing it can also be reconstructed by

(7)
• This is the fundamental relation for obtaining the impact force from the strain.
• once theLaplace transform components are obtained, thetime histories are obtained simply by using
theinverse Laplace transform. In this case, the force is obtained at the center of the beam (x =0).

COMPUTER IMPLEMENTATION
› A numerical Laplace transform algorithm can be used to
conveniently convert a time function into its frequency
components.
› The equations defining the Laplace transform and its inverse
transform with v(t) =0 for t<0 are

› Using the algorithm of the FFT, it is favourable to integrate
along lines parallel to the imaginary axis, i. e. ý, is constant, and
it follows that ds= idw

› Then the Laplace transform can be converted into the
Fourier transform

› where FT[..1] is the Fourier transform operator.

› For the evaluation of the Fourier integrals with the digital
computer, discrete Fourier transform is used.
› It results from the continuous formula if only a finite time
interval T is considered.
› This time internal is divided into N equaltime segments and
the integration is performed by using the Euler formula.

› the force history can be represented by

› where F*(wj)are the Laplace transform components at the
discrete frequencies wj
› The value of yT=6 has been used in this work. If only strain for the
forward wave is considered then Eqs. (6) and (7) can be written as
follows
›

:

CONCLUSIONS
› A new force-strain relation for the Bernoulli-Euler beam is
derived using the Laplace transform approach.
› The Laplace transform approach to dynamic strain
response analysis and impact force reconstruction
problems on an infinite beam can be simple, inexpensive
and very accurate.
› Operating on the force strain relation in the frequency
domain allows signals for arbitrary position x to be handled
much more conveniently than is the case in the time
domain.

fourier tranform application in euler and bernouili beam

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fourier tranform application in euler and bernouili beam