Identification of the dynamic characteristics of nonlinear structures

particular form of matrix perturbation analysis. Therefore, in order to guarantee
convergence of the iteration process, some restriction may have to be made on the
difference between the analytical and experimental models. This restriction is that the
Frobenius norms of the error matrices should be of second order when compared with
those of analytical mass and stiffness matrices (it is found that the convergence is largely
determined by the difference of these two models in the modal domain rather than the
spatial domain and the Frobenius norm is more closely related to the modal characteristics
of the system than any other norms), that is:
(6-12)

---

Identification of Mathematical Model of Dynamic Structures
183
Although varies from system to system, computational experience shows that for
dynamical systems, the maximum value of can reach 0.3. Since the norm is taken in its
Frobenius form and the modelling errors are usually 
the relative amplitudes of
modification for individual design variables can be more than 100% as shown in the
numerical case studies below.
6.4.2 
NUMERICAL CASE STUDIES
The first system studied here is an 
mass-spring system as shown in Fig.6.2. This
model has been used previously by several authors in model updating analysis exercises
and 
one of its noticeable features is that translational as well as rotational
motions are permitted, and that the mass matrix is not diagonal.
XC
XL
Fig.6.2 An 
System Used in 
Case Studies
When one complete column of the receptance matrix (column 1 in this numerical study)
has been measured, the solution of the updating problem is unique and the procedure
involved is direct since no approximation has been introduced in formulating the problem
as discussed in 
The analytical model is an undamped system with mass and
stiffness matrices shown in Tables 
The ‘experimental’ model consists, in
general, of mass, stiffness and damping matrices which are
= 
 
= 1.25 
and [D] = 0.02 
(6-13)

---

Identification of Mathematical Model of Dynamic Structures
184
Table 6.1 Analytical Mass Matrix (kg)
Table 6.2 Analytical Stiffness matrix (N/m)
Based on equation 
FRF data for 10 different frequency points (which were
randomly chosen in the ‘measured frequency range) were used each time to construct
[C(o)] and {B(o)) and as expected, the updated model is always exact regardless of the
number of modes which are included in the ‘measured frequency range (‘measured’ FRF
data covering the first mode, the first two modes and the first three modes were
considered and the mathematical proof will be given later to illustrate this point). Fig.6.3
shows the analytical, measured and regenerated (using the updated model) point
receptances at coordinate 
in the case the measured FRF data only covering the first
mode. As shown in Fig.6.3, not only the exact mass and stiffness matrices of the
‘experimental’ system are recovered, but also the exact damping matrix of the
‘experimental’ system.

---

6 Identification of Mathematical Model of Dynamic 
185
Fig.6.3 Analytical, Measured and Regenerated Point Receptances at Coordinate 
- - - - - analytical, 
measured and
regenerated
In order to consider the practical situation where the measured frequency response data
are contaminated by noise, 1% uniformly distributed noise was added to the ‘measured
frequency response function data and the above calculations were repeated. In all three
cases (‘measured’ FRF data covering the first mode, the first two modes and the first
three modes), it was found that the first iteration gave an estimation of errors to within
10% and after 2 or 3 iterations, the estimation error can be brought within 1% (for the
stiffness and mass matrices but not damping). No further improvement is possible
because of the presence of noise in the measured FRF data. The iteration strategy was
introduced because in this case of data contaminated by noise, the effect of noise on the
accuracy of estimation becomes less severe when the two models become closer. Again,
the analytical, measured and regenerated point receptances of coordinate are shown in
Fig.6.4, from which it is clear that the damping of the experimental system is
underestimated because the existence of noise has drowned the effects of damping on 
resonance frequency response data. This demonstrates the difficulties involved in the
damping property investigation of practical structures since the influence of damping on
the frequency response data (off resonance) is always of second order. This problem of
noise effecting the damping estimation can be overcome by smoothing the measured FRF
data first or by using the data points around system resonances during the updating
process.

---

6 Identification of Mathematical Model of Dynamic 
186
I
Fig.6.4 Analytical, Measured and Regenerated Point Receptances at Coordinate 
1% noise)
- - - - - analytical, 
regenerated
When the measured coordinates are incomplete, the updating procedure is the same as for
the case of complete coordinates except that during the formulation of equation (6-l the
unmeasured parts of the receptance 
are replaced by their analytical counterparts
In this numerical case study, 4 coordinates 
and are supposed to be
measured. The analytical model is the same as that shown in Tables 
and the
‘experimental’ model is the analytical one perturbed in such a way that 30% stiffness
modelling errors and 2% localised damping are introduced between coordinates 
and
as shown in Fig.6.5. The point receptances at coordinate for each of these two
models are shown in Fig.6.6. The mass matrix is assumed to be correct in this numerical
case study although the method is equally applicable for the case of mass matrix updating
as will be shown in the case study of GARTEUR structure in 
Stiffness error matrix
Damping error matrix
Fig.6.5 Exact Stiffness Error and Damping Matrices

---

Identification of Mathematical Model of Dynamic Structures
187
Fig.6.6 Analytica and Experimental Point Receptances of Coordinate xl
analytical, . . . . . . . . . . . . . . . . . . . experimental
The program flow-chart for solving equation (6-l 1) iteratively is illustrated in Fig.6.7. As
in the case of a complete set of measured coordinates, FRF data at 10 different frequency
points were used in each iteration. Again, data covering a frequency range of just the 
mode, the first two modes and the first three modes was investigated and it was found
that in all these three cases, the error for the estimation (for the stiffness matrix) was less
than 1% after 10 iterations. The iteration results for the case of data covering just the first
mode are presented in Fig.6.8. The convergence criterion was chosen to be the relative
norm (Euclidean) changes (see figure 6.7) of the stiffness and damping matrices of two
successive iterations and it can be shown mathematically that under this criterion, the
convergence is absolute (if it converges, it will converge to the true solution). As
expected, the damping matrix converges more slowly than the stiffness matrix and only
after the stiffness matrix has been obtained to some accuracy does the convergence of the
damping matrix becomes faster.

---

6 Identification of Mathematical 
Model of Dynamic Structures
188
complete analytical model
I
r
calculate [U( 
and {V(o)} based on (6-l 1) and
calculate 
condition number
of 
[solve [U(o)] {P) = 
and modify [M] [K] [H] 
= II K NH1 = 
II 
H 
I 
NK
 
solve eigenvalue problem in terms of 
[K] and
[H] and calculate [ for the next iteration
Iteration Failed
(reduce the number of unknowns
or measure more data)
Fig.6.7 Program Flowchart for Analytical Model Updating

---

6
of Mathematical Model of Dynamic 
189
Stiffness 
Matrix (first iteration)
Calculated Damping Matrix 
iteration)
Stiffness Error Matrix 
iteration)
Calculated Damping Matrix 
iteration)
Stiffness Error Matrix 
iteration)