Identification of the dynamic characteristics of nonlinear structures

 
Am, +
 
+
 
( 6 - 7 )
r = l
s = l
Here 
 
is the known coefficient vector formed using 
and [a,(o)] and 
is an unknown vector of design variable changes. Also, for a specific frequency value 
are known as the difference between analytical and experimental
receptance of coordinate (with excitation being applied at 
Therefore, from 
the following linear algebraic equations in terms of unknown design variable vector 
can be established
= 
where [C(o)] =
 
and (B(o)) =
In the presence of damping, [C(o)] and (B(o)) become complex, while {P) is known to
be real, and equation 
can be reformulated to solve for {P} as:
(6-8b)

---

Identification of Mathematical Model of Dynamic Structures
181
To illustrate how the coefficient matrix [C(o)] can be obtained, consider a simple 
mass and three-spring 2DOF system with its analytical mass and stiffness matrices as
follows:
(6-9)
Suppose the first column of
measured, then (6-6) becomes:
the receptance matrix of the ‘experimental’ system is
2
 
 
2
 
 
 
 
 
 
10)
 
 
 
 
 
 
where the frequency term has been dropped from the RHS of (6-10) to save space.
Comparing (6-10) with 
the coefficient matrix [C(o)] and vector (B(o)) are
obtained.
Equation (6-8a) (or 
in the case where damping exists) is obtained using the
analytical and measured receptance data at one frequency point, but when j frequency
points are used, then the total number of linear algebraic equations become j times as
many as that of using one frequency point and 
(or 
becomes a set of
overdetermined algebraic equations. In order to solve for (P) in this case, the best
technique available is the Singular Value Decomposition (SVD) which is described in
Appendix I. Since no approximation has been made during the formulation of the
problem, {P} can be solved directly. After 
is calculated, and together with the
analytical model itself, the updated system matrices can be determined uniquely. Also, it
should be noted that in this case, no assumption has been made concerning the
magnitudes of the error matrices [AM] and [AK].
The assumption that measurements are made in all the coordinates which are specified in
the analytical model is, in most cases, unrealistic because in many cases: certain
coordinates are physically inaccessible, such as the internal 
and the rotation
coordinates are notoriously difficult to measure. When the measured coordinates are

---

6
Identification of Mathematical Model of Dynamic Structures
182
incomplete, as will be discussed in Chapter 7, direct solution of the updating problem is,
in general, impossible and some sort of iteration scheme has to be introduced. Suppose
one 
incomplete 
column 
has been measured, then the multiplication of the RHS
of 
which requires the complete vector 
can no longer be carried out
exactly and some approximation has to be introduced.
Filling the unmeasured coordinates of
in (6-6) with their analytical counterparts
and then carrying out the multiplication of the RHS of (6-6) in the same way as for the
complete coordinate case leads to the following linear algebraic equations which are the
first order approximation:
(6-11)
where 
and 
are 
obtained in a similar way to [C(o)] and 
(B(o)) 
in the case
where measured coordinates are complete. However, in addition to the the approximate
nature of (6-l 
the total number of linear algebraic equations involved in (6-11) is n
when data one frequency point are considered while in 
the number is N 
where n is the number of measured coordinates and N is total number of degrees of
freedom specified in the analytical model). Again when more frequency points are used,
(6-l 1) can be turned into an overdetermined set and a least-squares method (SVD) can be
used to solve for {P). Of course, the thus obtained {P) is only the first order
approximation and an iteration scheme has to be introduced in order to obtain the exact
solution.
Also, it should be mentioned that in the case where the measured coordinates are
incomplete, the updating problem formulated in (6-l 1) is, mathematically, based on a