Estimination of unknown parameters in mechanical models for determination viscoelasticity properties of materials.

Obucina, Murco ; Bajramovic, Rasim ; Dzaferovic, Ejub 等

1. INTRODUCTION

Wood and wood products have elastic and viscous characteristics and belong to viscoelasticity materials (Dinwoodie, 2000). Viscoelasticity wood behaviour is very complex can be described by nonlinear models for which the unknown parameters are determined according to the experimental data. Faster convergence of nonlinear regression can be obtained by parameter grouping (Christensen, 1971).

2. RHEOLOGICAL MODELS AND EXPERIMENT

Creep test of laminated elements was carried out on a device with four-wheel loading (Figure 1). The device was placed in a chamber where relative humidity was maintained constant (93%) with axial fans which forced air flow to move over a dish with saturated solution of water and [K.sub.2]S[O.sub.4]. The room temperature was 22.5[degrees]C.

[FIGURE 1 OMITTED]

Constant loading was realized by weights (Figure 1 position 6). which is 17.5 % of the maximal loading.

At relatively low stress levels of humidity and temperature, wood may be considered as a linearly elastic material, and as a linearly viscoelastic material under others. Forms of the simplest viscoelastic models, which are used to represent the viscoelastic behavior of wood materials, are the Standard linear model Equation (1) and Burger model Equation (2) (Skrypek & Hetnarski 1993):

J(t) = [1/[E.sub.1] + 1/[E.sub.2] - 1/[E.sub.2] exp (-[tE.sub.2]/[mu])] (1)

J(t) = [1/[E.sub.1] + 1/[E.sub.2] (1 - exp(-[tE.sub.2]/[[mu].sub.2])) + t/[[mu].sub.1]] (2)

where J(t) is the creep compliance; t is the time; [E.sub.i] is elastic constant for springs; and [[mu].sub.i] is viscous constant for dashpots (i = 1 and 2). According to Figure 1. the elastic deflection at middle point is (Obucina et al., 2006).

y = 23(F/2)[l.sup.3]/648EI (3)

Where: F is the force ; / is the reference length; I is the moment of inertia of the beam's cross section and E is the elastic modulus.

If a viscoelastic beam is subjected to the same force, as in the elastic case, the deflection can be derived multiplying Equation (4) by E(t)/E, where E(t) is the relaxation modulus. If 1/E(t) is replaced with J(t), where J(t) is the creep compliance, Equation (4) becomes

y(t) = K x J(t); K = -23/648 (F/2) x [l.sup.3]/I (4)

To determine the mechanism of any physical process it is necessary to establish a correspondence between experimental observations on the one hand and the mathematical model on the other. Any model can be written in the form

[y.sub.i] = f([x.sub.i], [theta]) + [[epsilon].sub.i] i = 1, 2, ..., n (5)

Where: -[y.sub.i] is the value of the dependent variable at the i-th measurement

--[X.sub.i] = ([x.sub.1i], [x.sub.2i], ..., [x.sub.qi])' is a q-dimensional vector of independent variables at the i-th trial

-[theta] = ([[theta].sub.1], [[theta].sub.2], ..., [[theta].sub.p])' is a p-dimensional vector of unknown parameters. -[[epsilon].sub.i] is an experimental error at the i-th trial.

Empirical models, which are usually polynomials in the independent variables but linear in the parameters, are usually used to predict for purposes of optimization and control of processes Another type of models is mechanistic model based on physical mechanism of the observed process. The models with four and three parameters used in this work are nonlinear.

In matrix notation the relationship between the dependent variable and the independent variables for models linear in parameters can be written as (Draper & Smith 1988).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The vector of unknown parameters can be estimated by the method of least squares as

[??] = [(X'X).sup.-1] X'y (7)

Equation (7) can be solved if the matrix X'X is nonsingular. If this matrix is singular then vector of parameters is not uniquely defined.

With assumption that errors [epsilon] have normal distribution with constant variance-covariance matrix of unkown parameters is defined as

V[??] = [(X'X).sup.-1] [[sigma].sup.2] (8)

It often may be assumed for nonlinear models that the dependent variable can be adequately represented over some interval in the parameters by a truncated (linear) Taylor series expansion about preliminary estimates of unknown parameters. With this assumption the nonlinear model can be written approximately in the form

z = X([theta] - [??])+ [epsilon] (9)

Where: z = [[y.sub.i] - f([x.sub.i], [??])] i-th element (10)

X = [[[partial derivative]f([x.sub.i], [theta])/[partial derivative][[theta].sub.j].sub.[theta] = [??]]

[??] is a vector of the numerical values about which the linearization is made.

Equation (9) represents the approximate model linear in parameters and as before the unknown parameters can be estimated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

and residual sum of squares is given as R = z'z

The iterative procedure is terminated when the correction obtained by (12) becomes small enough; i.e. the change in parameters is insignificant. According to the given procedure software made in [C.sup.++] computer language was made. For faster convergence "the doubling and halving technique" and optimization of parameter vector change were applied at each iteration step.

3. RESULTS

In order to compare two models F-test was carried out for the results given and the calculated results are given by curves in Figure 3. For 4- parameters model estimated variance is [s.sup.2.sub.4] = 2.20360e-02 with 99 degrees of freedom and for 3-parameters model variance is [s.sup.2.sub.3] = 1.02391e-01 with 100 degrees of freedom. The corresponding F value is F = [s.sup.2.sub.3]/[s.sup.2.sub.4] = 4,6465. Since this value is greater than table values for both [alpha] = 0,05 and [alpha] = 0,01 ([F.sub.0.05,100,99] = 1.39 and [F.sub.0.01,100,99] = 1.60), it means that 4-parameters model is significantly better than 3 parameters model.

[FIGURE 3 OMITTED]

In order to improve convergence of nonlinear regression method the grouping of parameters can be performed. For this case 4-parameters model was considered. This model can be transformed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where a = -1/[E.sub.2]; b = [E.sub.2]/[[mu].sub.2]; c = 1/[[mu].sub.1]; i d = 1/[E.sub.1] +1/[E.sub.2]

According to the nonlinear regression method the following results were obtained.

[E.sub.1] = 13701; [E.sub.2] = 11658; [[mu].sub.1] = 420594; [[mu].sub.2] = 28779; R = 4.165 a = -8.57765 x [10.sup.-5]; b = 0.405083; c = 2.37754 x [10.sup.-6]; d = 1.58763; R = 4.165

Transforming a, b, c and d the next values are obtained [E.sub.1t] = 13703; [E.sub.2t] = 11658; [[mu].sub.1t] = 420602; [[mu].sub.2t] = 28780; R = 4.165.

4. CONCLUSION

According to the given F-test 4-parameters model is significantly better them 3-parameters model. The convergence was faster for transformed model obtained by parameter grouping. Obtained results are the same for both cases. The problem will appear if we want to determine confidence intervals for parameters. There is no problem for original model since parameters have physical meaning. In the case of transformed model confidence intervals for parameters a, b, c and d can not be transformed into confidence intervals for [E.sub.1]; [E.sub.2]; [[mu].sub.1]; [[mu].sub.2]. If we need models only for prediction then it is useful to use transformed model, but in the case that we need confidence intervals for real parameters then original model should be considered no matter how difficult is to estimate it.

5. REFERENCES

Christensen, R. M. (1971). Theory of Viscoelasticity an Introduction. Academic Press, 0-12-174250-4, New York

Dinwoodie J. M. 2000. Timber its nature and behavior, 0-41925550-8, London and New York.

Draper, N.R. and Smith, H. (1988). Applied Regression Analysis. Wiley.

Obucina, M.; Dzaferovic, E.; Bajramovic, R. & Resnik, J. (2006). Influence of gluing technology on viscoelasticity propert of LVL, Wood Research, 51(4), 11-22 1336-4561

Skrzypek J.J., Hetnarski R.B. (1993) Plasticity and Creep, theory, examples, and Problems, International Standard Book Number 0-8493-9936-X.