4.3 Model Calibration
4.3.10 Application: Energy Dissipation in a Lap Joint
Sections 4.3.1 - 4.3.9 discussed several aspects of model calibration and parame- ter estimation. Now, an engineering application problem in the area of mechanical engineering is chosen for illustrating the proposed methods. For the sake completion, calibration is performed using both classical statistics-based methods and likelihood- based Bayesian methods.
4.3.10.1 Description of the Problem
This example deals with the calibration of the Smallwood model [91, 133, 134], which is used to predict the energy dissipation due to friction at a lap joint in a mechanical component. This model predicts the dissipation energy (DE) per cycle at the joint when the component is subjected to an impact harmonic force of amplitude F.
The hysteresis curve (force vs. displacement graph) for the lap joint comprises of two symmetrical portions. The energy loss in the joint under one cycle of sinusoidal loading is found by integrating the area under the hysteresis curve and analytically derived as:
DE =kn(m−1
m+ 1)zm+1 (4.25)
In Eq. 4.25, kn is a non-linear stiffness term, m is the exponent term, and z is the displacement amplitude which is obtained by solving:
2F =kz−knzm (4.26)
In Eq. 4.26, k refers to a linear stiffness term. The objective is to calibrate the non- linear stiffness parameter (kn) using the available input-output data. Data is available on the inputs - force (F), linear stiffness (k), and non-linear exponent (m) - and the
output - dissipated energy (DE). For a given force, the dissipated energy can be measured, and hence there is correspondence and ordered pairing between the input force (F) and output energy (DE). There are five such measurements as shown in Table 4.1.
Table 4.1: Calibration Data: Force vs. Dissipated Energy
Force (F) Dissipated Energy (Z) (inlbf) (inlbf×in)
60 5.30×10−5 120 2.85×10−4 180 7.78×10−4 240 1.55×10−3 320 2.50×10−3
LetFj (j = 1 to 5) andZj (j = 1 to 5) denote the five force values and five energy values in Table 4.1. A different symbol Z has been used for the output measurement in order to avoid confusion with the symbol DE used for model prediction. The variables Z and DE are related through the fitting error (ǫ∼N(0, σ2)), as:
f(Z|DE) = 1 σp
(2π) exp−
(Z−DE)2 2σ2
(4.27)
Two other inputs, linear stiffness (k) and non-linear exponent (m) are not mea- sured in correspondence with the force measurement and hence are not paired with the output measurement as well. For the sake of illustrating the methods, it is assumed that this unpaired data has come from other independent sources (other experiments, subject matter experts, etc.)
The data on linear stiffness (k) is available as: three intervals ([1160000, 1180000], [1155000, 1170000], [1160000, 1170000]) and one point value (1173000). All measure- ments are in lbf /in.
The non-linear exponent (m) is known to have a normal distribution with mean
= 1.23 (no units) and coefficient of variation = 0.06. Thus, this numerical example
features several types of uncertainty - (1) additional source of uncertainty (m); (2) interval data (k); and (3) uncharacterized data (k).
For the purpose of model calibration, Eq. 4.25 and Eq. 4.26 can be expressed together as
DE =G(F, k, m;kn) (4.28) Eq. 4.28 is obtained by eliminating z from Eq. 4.25 and Eq. 4.26. The various steps in the calibration procedure are:
1. Represent each unpaired input using a PDF. There are two unpaired inputs.
The linear stiffness (k) is given by three intervals and one point value, and the nonlinear exponent (m) is known to be normal. Thus, there is a well-defined PDF fm(m) in the latter case whereas the PDFfk(k) needs to be constructed for the former case.
2. Once fm(m) and fk(k) are known, then the model in Eq. 4.28 needs to be evaluated for each value of the paired input, i.e. for each of the force values in Table 4.1.
3. In the least squares approach, an error measure is computed as a function of the calibration parameter (kn) and minimized whereas in the likelihood ap- proach, the likelihood function of the calibration parameter (kn) is computed and maximized.
4. The uncertainty in least squares estimation is expressed through confidence intervals, and the likelihood function is used in Bayesian inference to calculate the entire PDF of the calibration parameter (kn).
These four steps are explained in detail below.
4.3.10.2 Least Squares Approach
In the first step, fm(m) is known to be a normal density function with mean = 1.36 and coefficient of variation = 0.05.
The PDF fk(k) is constructed as a composite PDF - a weighted sum of three uniform PDFs ([1160000, 1180000], [1155000, 1170000], [1160000, 1170000]) and one Dirac delta PDF centered at the available point data (1173000). Uniform weights are used for each of these 4 data; hence each of these data is assigned a probability of 0.25, and cumulative probabilities of 0.25, 0.50, 0.75, and 1 respectively. The algorithm for generating samples from this composite PDF is:
1. Generate random number. Call it “temp”.
2. If temp <0.25, Generate a sample fork from the uniform distribution bounded on [1160000, 1180000]. End. To generate next sample, Go to Step 1.
3. Else if temp < 0.50, Generate a sample for k from the uniform distribution bounded on [1155000, 1170000]. End. To generate next sample, Go to Step 1.
4. Else if temp < 0.75, Generate a sample for k from the uniform distribution bounded on [1160000, 1170000]. End. To generate next sample, Go to Step 1.
5. Else, the sample for k is 1173000. End. To generate next sample, Go to Step 1.
Using the above generated samples, the PDF of k is generated and shown in Fig. 4.3.
This completes the first step.
In the second step, the PDF of the model prediction is estimated as a function of Fj (j = 1 to 5) andkn. It needs to be a function ofF because the input measurement (F) and the output measurement (Z) have a one-one correspondence, as in Table 4.1.
It needs to be a function of kn because this quantity is the calibration parameter
1.15 1.16 1.17 1.18 1.19 x 106 0
0.2 0.4 0.6 0.8 1 1.2 1.4x 10−4
Linear Stiffness (in N/m)
Figure 4.3: Composite PDF of Linear Stiffness
and the squared error measure should be a function of kn. Uncertainty propagation methods (Section 2.5) are then used to compute the PDF of model prediction for each ordered input measurement; this PDF is denoted as fDEj(DEj|Fj, kn). In the third step, this PDFfDEj(DEj|Fj, kn) is compared (so as to compute an error measure Sj(kn)) with the corresponding output measurement data denoted by Zj, which is a point value.
The comparison of a PDF with a point value is not straightforward; this PDF fDEj(DEj|Fj, kn) cannot be effectively used in the least squares approach, and this is one reason why the least squares approach is not used in the remainder of this dissertation. Nevertheless, for the sake of completeness, this example problem is completed as follows.
The error metric S(kn) is computed by measuring the difference between the measurement Zj and the expectation of the model prediction, i.e.E(DEj|Fj, kn), as:
S(kn) =
5
X
j=1
Sj(kn) =
5
X
j=1
E(DEj|Fj, kn)−Zj
2
(4.29)
(If the output measurement were an interval instead of a point value, then a uniform PDF can be considered on this interval and this uniform PDF is compared with fDEj(DEj|Fj, kn) to compute Sj(kn). A distance metric such as the area met- ric [135], Kullback-Leibler Divergence [136], Hellinger distance [137], Bhattacharyya distance [138] etc. can be used to compute the “distance” or “difference” between two PDFs. It must be noted that the choice of a uniform distribution is an addi- tional assumption. In the likelihood/Bayesian approach, these difficulties are easily overcome using the inherent definition of likelihood.)
This error metric S(kn) is minimized and the calibration parameter (kn) is esti- mated. Further, the uncertainty in this estimate is calculated using the F-statistic, as explained earlier in Section 4.3.2. The least squares estimate is found to be 6.62 x 105lbf /inand the 95% confidence interval is calculated to be [6.59 x 105lbf /in, 6.66 x 105lbf /in].
4.3.10.3 Likelihood Approach
In the likelihood approach, the first step and the third step are different from the least squares approach; the second step of computing the PDFfDEj(DEj|Fj, kn) using uncertainty propagation is the same.
In the first step, the PDFfk(k) is computed using the non-parametric likelihood- based technique described earlier in Section 3.7, and the resulting PDF is shown in Fig. 4.4.
This PDF is estimated by solving the optimization in Eq. 3.30; the domain of [1155000, 1180000] is discretized into 10 equal parts, and the density values at these discretization points are estimated so that the likelihood for the given data (3 intervals and one point) is maximum. Note that this is a non-parametric PDF constructed using the Gaussian process interpolation method based on the PDF values at the
1.155 1.16 1.165 1.17 1.175 1.18 x 106 0
0.2 0.4 0.6 0.8 1 1.2 1.4x 10−4
Linear Stiffness (in N/m)
Figure 4.4: Likelihood-based PDF of Linear Stiffness (k)
10 discretization points. Also, this PDF does not make an additional assumption (uniform distribution within each interval which was assumed in Fig. 4.3). This completes the first step in model calibration.
As mentioned earlier, the second step is the same as in the least squares ap- proach, and the PDFs fDEj(DEj|Fj, kn) (j = 1 to 5) are calculated using uncertainty propagation.
In the third step the likelihood function is calculated using the methods developed in Section 4.3.8. Then, this likelihood function can be maximized; further the uncer- tainty in this estimate can be calculated using Bayes theorem as explained earlier in Eq. 2.7. A non-informative uniform distribution for the prior ofkn, and the resulting posterior PDF is shown in Fig 4.5.
The maximum likelihood estimate is 660690 lbf /in; the 95% probability bounds are given by [617160 lbf /in, 689060lbf /in].
5.2 5.6 6 6.4 6.8 7.2 7.6 x 105 0
0.5 1 1.5 2 2.5x 10−5
Nonlinear Stiffness (in N/m)
Figure 4.5: Estimated PDF of Nonlinear Stiffness (kn)
4.3.10.4 Discussion
To begin with, it is acknowledged that there is no single correct answer to the model calibration problem with imprecise and unpaired input-output descriptions.
First, there are several procedures in the literature to treat interval data and each procedure may lead to a different answer. Ferson et al. [97] discuss in detail several aspects of interval data treatment. The least squares procedure is able to address interval data only by assuming a uniform distribution within each interval. As stated earlier, this assumption is questionable. The likelihood approach does not make any such assumption and hence, is faithful to the data, as far as possible. Second, the issue of correspondence between input and output measurements directly affects the uncertainty in the calibrated quantity. Note that the multi-modal behavior of the input k is reflected in the PDF of the calibrated kn in Fig. 4.5.
Also, the estimates of uncertainty obtained from the least-squares based approach and the likelihood approach need to be interpreted differently. The likelihood ap- proach alone can provide the entire PDF of the calibrated kn and the least squares
approach cannot; it only provides confidence intervals which need to be interpreted differently from probability bounds [9]. A 95% confidence interval for a model param- eter is interpreted as follows: “Suppose that the data collection process was repeated 100 times. Hence, 100 different least squares estimates, and 100 corresponding confi- dence intervals can be calculated. Then, 95 out of 100 intervals will contain thetrue estimate of the model parameter”.