## Beamlsd

In this subsection the one superelement approximation of an Euler-Bernoulli beam is used to generate the "measured" data. This model is referred to as Beamlsd and is depicted in Fig. 5.4. The Euler-Bernoulli beam considered has length L = 50 m, modulus of elasticity E = 21 • 1010 N/m2, area of beam cross-section (circular) A = n m2, mass density p = 7850 kg/m3, and area moment of inertia I = 4n m4. The beam is built in at the base.

Euler-Bernoulli beam

2 Ur

One superelement approximation

Figure 5.4: One superelement approximation of an Euler-Bernoulli beam.

The torsional spring constants for the superelement can be derived directly from the data mentioned above using the automated structural modeling procedure outlined in Section 3.4. The resulting constants are

Viscous damping, sufficient to produce a damping ratio of 1 %, has been added to the model by specifying the following coefficients of viscous damping

Keq 12 = 3.70 • 107 [kg/s] Keq23 = 3.25 • 106 [kg/s]

It is assumed that the mass, inertia and length of the rigid bodies within the superelement are estimated fairly accurately, but that calculation of the stiffness and damping parameters may be difficult. Consequently, Cz1, Keq12, Cz3, and Keq23 are the tunable parameters. The aforementioned values of the spring and damper constants form the real parameter vector d0. Thus in this case d = [ Cz1 Keq 12 Cz3 Keq23 ]

and consequently d0 = [ 6.5973 • 109 3.70 • 107 6.5973 • 109 3.25 • 106 ]T

The input is a stepwise change in the force at the beam top (step time 0 s, initial value of zero and final value of 80000 N) producing a maximum deflection of 0.04 m. The output is the acceleration of the beam top. Time histories of 301 points are generated using the Runge-Kutta fourth order (RK-45) method to numerically integrate the differential equations with a fixed step size of 0.05 s. The step size is selected such that at there are least three points within the smallest oscillation period. Furthermore, it has been checked that halving the step size has no effect on the results.

The objective function is defined as the sum of squares of the difference between the measured and simulated output. The objective is depicted in Fig. 5.5 as a function of the percentage of variation in the parameter vector 0 (i.e. 100% = 00, and 110% means that the numerical values of all four tunable parameters are set to 110 % of their real values). Observe the two local minima around 75 % and 125 % of the real parameter vector.

Objective function

Objective function

Figure 5.5: Objective function Fobj of Beamlsd with o calculated values.

Figure 5.5: Objective function Fobj of Beamlsd with o calculated values.

It is now interesting to determine which parameter variations around the real values of Cz1, Keq12, Cz3, and Keq23 can be allowed such that the estimated parameter vector 0 still converges to the real parameter vector 00. That is, determine max {A0o} s.t. 0 ^ 0o

Two noise-free cases and one where a white-noise signal has been added to represent measurement noise are considered in the sequel. The first and third case use the Nelder-Mead simplex method implemented by the fmins function in the Optimization Toolbox of MATLAB®. The second uses a combined Gauss-Newton and modified Newton algorithm to minimize the unconstrained sum of squares. The algorithm is the so-called e04fdf routine from the Numerical Algorithms Group (NAG) Toolbox [197]. Both algorithms use function values only.

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