Abstract

Identification of Flexural Stiffness Parameters of Beams

José João de Espíndola

Department of Mechanical Engineering

Federal University of Santa Catarina. Brazil

espindol@mbox1.ufsc.br

João Morais da Silva Neto

Department of Mechanical Engineering

Federal University of Santa Catarina. Brazil

joaoneto@emc.ufsc.br

Introduction

The cross section stiffness of beams of complex constructions, such as sandwich beams and stranded cables of the type ACSR, are of difficult evaluation by analytical and/or numerical means.

Also of difficult assessment is the inherent damping of such constructions. In many instances the damping of ACSR cables, for example, is measured by the logarithmic decrement of a segment of the cable, fixed at both ends, under a certain mechanical tension. This practice is conceptually wrong, for the logarithmic decrement is not a measure of the inherent damping of the "material", that is, of the inherent damping of the cable (Lazan, B. J., 1968). Rather, it is a measure of the "system" formed by the piece of cable fixed at both ends. For a different span, different logarithmic decrement is bound to be measured. Logarithmic decrement is a structural parameter, not a material one.

In this paper the beam, no matter if a piece of ACSR cable, or a multi-layered sandwich beam, is modelled as an equivalent continuous viscoelastic beam of complex cross section flexural stiffness (ECFS) equals to , where is the effective cross section flexural stiffness and h is the equivalent loss factor, i. e., the loss factor of the viscoelastic material from which the beam model is made. The purpose of this paper is to introduce the concept of equivalent complex cross section flexural stiffness (ECFS) for beams and to produce an approach to measure, or identify it. To achieve this objective a built up structure , or test sample, is constructed and tested experimentally. The beam from which the ECFS is desired is part of this built up structure, or test sample (see fig.1). This built up structure closely resembles a symmetric Stockbridge neutraliser (Teixeira. P.H.,1997). It is stressed here, although, that this paper is not at all concerned with, nor about Stockbridge neutralisers. As a matter of fact, it is not about neutralisers of any sort. The above resemblance is noted "en passant", and is supposed to clarify, not to confound. Note: In this, as well as in all the previous papers of the first author, the name "dynamic vibration neutraliser", or DVN, is used instead of "dynamic vibration absorber", or DVA, simply because this device is not an absorber, or damper at all. The words "absorber" and "damper" are inaccurate (Crede, C.E.,1965).

Once the ECFS is known, both the real flexural stiffness (EI)_r and the effective loss factor h of the "material" (for instance, a stranded cable) are also known. These data may be precious for many practical applications, such as the design of overhead power transmission lines and other cable structures.

In beams made up of alternate layers of metal and viscoelastic material, both and h may be heavily dependent on frequency (Snowdon, J. C., 1968). Although this dependence is currently being investigated by the authors, it is not taken into account in this paper. Hereby, both and h are not supposed to vary with frequency.

The parameters and h (or and ) are measured, or identified, by fitting a theoretical model of the built up beam structure (test sample) to a FRF measured at the root of it (Silva Neto, J. M., 1999).

Three Degree of Freedom Model of the Test Sample

Fig. 1 shows half the symmetric beam structure, or test sample. The test sample is build up with a rigid body at one end of the beam and is excited at a central mass by a force f(t). It is assumed no rotation at the fixed section at the central mass, thanks to symmetry. In actual practice, some lack of symmetry is unavoidable. The experiment must then allow for some means of preventing such rotations.

The beam itself is assumed to have no mass.

The stiffness matrix in the co-ordinates and can easily be shown to be (Espindola, J. J., 1987):

where the semicolon means change of line. This is a complex stiffness matrix, since (the ECFS) is a complex number. h is the loss factor of the viscoelastic material of the beam model.

The quantities shown in fig. 1 are:

m_c - mass of the rigid body of the test sample;

J_c - moment of inertia of the above rigid body in relation to a baricentric axis, normal the plane of the paper;

e - distance from the free end of the beam element to the mass centre of the rigid body;

y(t) - displacement of the root of the beam structure;

f(t) - exciting force.

The kinetic and potential energies for half the test sample model are:

and

where the dot over quantities mean derivative in relation to time.

Substitution of (2) and (3) into the Lagrange Equations (Meirovitch, L., 1970, 1990) results:

This result may be written in the following form:

Fourier transforming both sides of (5) and after a few algebraic manipulations, one gets:

In the above expression, one has as the mass matrix in the co-ordinates and and . M(W) is the dynamic mass at the root of the test sample.

An alternative expression that has proved to be more convenient to work with is given below, easily derived from (6):

where F is modal matrix (real) and L, the spectral matrix (complex) of the eigenvalue problem _{Kf = lMf}. The modal matrix is constructed in such a way that its columns are the eigenvectors of _{Kf = lMf} Note that K is also complex.

The spectral matrix is a diagonal one containing the eigenvalues of the above eigenvalue problem. Expression (7) assumes that the eigenvectors are orthonormal in relation to the mass matrix, that is, that they obey the relation

In such a condition one has:

The eigenvalues in the diagonal of L may be written in the form:

In this expression , j=1,2, is the j^th undamped natural frequency and h_j is the corresponding modal loss factor of the test sample. The modal loss factors are structural parameters of the whole test sample, not material parameters of the beam segment. So they must not be confused with h in the expression .

The Cost Function

At this stage a cost function must be defined so that the FRF of expression (7) be made as close as possible to its measured counterpart. This is accomplished by varying the parameters and contained in (7). This approximation of mathematical model (7) to its measured counterpart may be made through an appropriate nonlinear optimisation algorithm or a genetic or even a hybrid algorithm (Espindola and Bavastri, 1997, 1998). The error function is defined by taking the difference, for each value of W, between mathematical model (7) and its measured counterpart:

where M(W) is given by (7) and is the measured FRF.

Note that E(W) is a complex-valued function defined on the real number set.

The cost function may be defined as follows:

where the asterisk * stands for complex conjugate and x is a design vector given by:

In the above expression, is the real part and is the imaginary part of EI, so that h = (EI)_r/(EI)_i. R_r and R_i are the real and imaginary parts of a residual term to be added to the mathematical model (7) to take into account effectively existing modes above the measurement base-band (Ewins, D. J, 1984, Natke, H. G. and Cottin, N.,1988, Maia, N.M.M. and Silva, J.M.M., 1997).

In vector (13), e and J_c are taken as design variables to allow them to account for some inertial effects due to actual mass of the beam. Also, in the case of a Stockbridge neutraliser, e and J_c are difficult to measure accurately due to the shape of the end masses .

Finally, a complementary equation is necessary so that M(W), in equation (7), can be effectively computed. This equation is precisely the eigenvalue problem _{Kf = lMf} .

Measurement Setup

Figure 2 shows a Stockbridge neutraliser (made by Wetzel, SC, Brazil, model AS-2008) which was effectively taken as one particular test sample. This neutraliser was taken as a test sample only because it was readily available. The purpose of this particular experiment was then to find the ECFS of the stranded cable, part of this Stockbridge neutraliser. In the end both (EI)_r and h for this cable would be available. The value of this h is an important parameter to assess the efficiency (or lack of) of this sort of neutraliser in reducing overhead transmission line vibrations over a wide frequency band (Espindola ,J.J. and Bavastri,C.A., 1997 and Espíndola,J.J.,et al.,1998).

The Stockbridge test sample was suspended by thin flexible cables at the central mass and excited horizontally with an electrical shaker. The acceleration response was measured at the same mass. Both signals were fed into a Dynamic Signal Analyser (DSA) and the FRF computed (see figure 3). The force signal was a swept-sine function. Frequency resolution was 0.3Hz. The measurement band ranged from zero up to three hundred Hertz (base-band). Mathematical model (7) was then fitted to the measured FRF and the cost function minimised by the DFP (Davidon-Fletcher-Powell) algorithm (Rao,S.S.,1996). After the minimisation is accomplished, the parameters in (13) are available.

Experimental Results

Table 1 gives the results of such an identification. In table 1, and .

Figure 4 shows a comparison between the measured FRF and the regenerated one. The agreement seems to be excellent. This is best shown by zooming both curves around the resonant picks as shown in figure 5, a and b.

To further boost the confidence in the above approach, another test sample was designed and constructed, with a cylindrical beam element of diameter d made out of hot rolled carbon steel, with (EI)_r known in advance. Figure 6 is a photo of that test sample. Figure 7 shows details of the end masses.

The data for this particular test sample are shown in table 2. The beam was fixed at the end masses by shrink fit.

Table 3 shows the numerical results after identification.

If a comparison is made of the value of (EI)_r in table 3 with EI in table 2 a conclusion is drawn that they differ by only 1.7 per cent. The loss factor of the material (low carbon steel) is computed from (EI)_r and (EI)_i , giving 2.97x10^-4, which is within the expected range of 2x10^-4 to 6x10^-4 for steels (Rao, S.S.,1995).

One can see that the three degree of freedom model of the test sample presented here is well adequate for the purpose of identifying the ECFS of beams and, as a consequence, its effective flexural stiffness and inherent damping .

A comment on the inherent damping of Stockbridge neutralisers is at order, in spite of the fact that this paper is not at all about those devices. The loss factor of 0.011, table 1, is very low indeed. This explains why such a neutraliser is so efficient in reducing vibrations in a very narrow band of frequencies around its first natural frequency, but so poor in mitigating vibrations outside that band. Since the band of frequencies exciting overhead lines is actually much wider, far more damping is needed in a neutraliser than that provided by the Stockbridge one. The first author has designed a neutraliser with a great amount of viscoelastic damping and excellent performance over a wide band of frequencies and patented it (Espindola,J.J. et al,1998).

Final Remarks and Conclusions

A three degree of freedom model of a test sample resembling a Stockbridge neutraliser was presented. The primary objective of it is the identification of the ECFS of the segment of beam, part of it, thus permitting the computation of its actual bending stiffness (EI)_r and equivalent loss factor.

One immediate application is the identification of these two parameters for stranded cables, such as those used in overhead transmission lines and other cable structures.

The results from two test samples, one Stockbridge neutraliser and one specially designed and built to check the approach, have given very encouraging results.

This technique is now being extended to sandwich beams, where the core is a very damped viscoelastic material. This poses an additional challenge, for very damped viscoelastic materials have their dynamic properties largely dependent on frequencies. This means that the ECFS for such beams must be identified in a band of frequencies. This technique, once developed, can be used to identify the dynamic properties of the viscoelastic core material over a band of frequencies.

Manuscript received: July 2000, Technical Editor: José Roberto de França Arruda.

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