System condensations for inverse problems of linear dynamic structures

Figures & data

Figure 1. Cantilever beam in flexural vibration.

Figure 2. Element modifications of perturbed system.

Table 1. Eigenvalues of baseline and perturbed systems

	Eigenvalues
Mode	Baseline	Perturbed	% change
1	2.80703 × 10^2	2.93392 × 10^2	+4.52
2	1.10244 × 10^4	1.14945 × 10^4	+4.26
3	8.64349 × 10^4	9.02208 × 10^4	+4.38
4	3.31948 × 10^5	3.47170 × 10^5	+4.59
5	9.07265 × 10^5	9.39662 × 10^5	+3.57

Figure 3. Selected primary sets.

Table 2. Convergence characteristics of selected primary sets

Scheme	λs	μ(S)
(A) Sequential elimination	1.6052 × 10^7	3.1859
(B) Linear independence	6.0858 × 10^6	3.4001
(C) Condition number	3.6485 × 10^6	1.6336

Figure 4. Solution errors in static condensation.

Figure 5. Element identification with exact transformation (dynamic stiffness matrix).

Figure 6. Element identification with exact transformation (modal matrix projection).

Table 3. Eigenvalues with exact transformation

			Inverse system
Mode	Perturbed system		Dynamic stiffness matrix		Modal matrix projection
1	2.93392 × 102	+4.52%	2.93303 × 10^2	+4.49%	2.93283 × 10^2	+4.48%
2	1.14945 × 10^4	+4.26%	1.14940 × 10^4	+4.26%	1.14955 × 10^4	+4.27%
3	9.02208 × 10^4	+4.38%	9.02473 × 10^4	+4.41%	9.02604 × 10^4	+4.43%
4	3.47170 × 10^5	+4.59%	3.47267 × 10^5	+4.61%	3.47200 × 10^5	+4.59%
5	9.39662 × 10^5	+3.57%	9.39922 × 10^5	+3.60%	9.39607 × 10^5	+3.56%

Figure 7. Inverse solution (dynamic stiffness matrix).

Figure 8. Inverse solution (modal matrix projection).

Table 4. Solution accuracy in eigenvalues

			Inverse system
Mode	Perturbed system		Dynamic stiffness matrix		Modal matrix projection
1	2.93392 × 10^2	+4.52%	2.93394 × 10^2	+4.52%	2.93384 × 10^2	+4.52%
2	1.14945 × 10^4	+4.26%	1.14946 × 10^4	+4.27%	1.15094 × 10^4	+4.40%
3	9.02208 × 10^4	+4.38%	9.02208 × 10^4	+4.38%	9.05601 × 10^4	+4.77%
4	3.47170 × 10^5	+4.59%	3.47171 × 10^5	+4.59%	3.47414 × 10^5	+4.66%
5	9.39662 × 10^5	+3.57%	9.39666 × 10^5	+3.57%	9.39919 × 10^5	+3.60%