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 Bernoulli-Euler 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

Bernoulli-Euler type.

› The applied force and the beam displacement are related by

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

(1)

FORCE-STRAIN RELATION› 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

FORCE-STRAIN RELATION

› 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.

FORCE-STRAIN RELATION

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

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.

(2)

FORCE-STRAIN RELATION

› 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)

FORCE-STRAIN RELATION› Imposing continuity of displacement and slope at

the joint, and writing the equations of motion of the joint itself gives (at x=0)

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

(5)

(6)

FORCE-STRAIN RELATION› 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

• 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).

(7)

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

COMPUTER IMPLEMENTATION

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

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

COMPUTER IMPLEMENTATION

› 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.

COMPUTER IMPLEMENTATION

› 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.

Thank you