Identification of mass, damping and stiffness matrices of multi degree of freedom system subjected to kinematic excitations

Majcher, Krzysztof

Identification of mass, damping and stiffness matrices of multi degree of freedom system subjected to kinematic excitations

Studia Geotechnica et Mechanica

Volume 47 (2025): Issue 1 (March 2025)

By:

Krzysztof Majcher

Open Access

|Mar 2025

Figures & Tables

Diagrams of applied kinematic excitations: a) harmonics, b) sum of three harmonic functions and c) real earthquake.

Spectral power density determined from displacements registered during 2006 Kiholo Bay earthquake.

Dynamic response of computational model at time instant t = 7_71s_

Iteration step	Displacements [mm]			Velocities [mm/s]			Accelerations [mm/s²]

	q₁	q₂	q₃	q˙1 {\dot q_1}	q˙2 {\dot q_2}	q˙3 {\dot q_3}	q¨1 {\ddot q_1}	q¨2 {\ddot q_2}	q¨3 {\ddot q_3}	z¨ \ddot z
1	5.039	7.062	9.491	17.36	23.66	13.55	−1274	−605.3	−933.9	648.8
2	4.324	6.151	8.897	25.15	22.09	14.97	−1172	−498.1	−965.6	648.8
3	3.508	5.281	8.293	31.56	19.94	16.40	−1025	−432.6	−991.9	648.8
4	2.614	4.444	7.685	35.78	17.88	17.78	−843.2	−398.6	−1014	648.8
5	1.681	3.623	7.079	37.28	16.25	19.11	−645.9	−379.1	−1034	648.8
6	0.754	2.798	6.479	36.02	14.96	20.44	−454.0	−356.7	−1056	648.8
7	−0.122	1.960	5.888	32.30	13.60	21.88	−285.5	−318.9	−1079	648.8

Computational model data in particular iteration steps_

Iteration step	1 ref.	2	3	4	5	6	7
m₁ [kg]	10.724
m₂ [kg]	10.134
m₃ [kg]	20.32
Δm₁ [kg]	0	0.1	0.2	0.3	0.4	0.5	0.6
Δm₂ [kg]	0	0.1	0.2	0.3	0.4	0.5	0.6
Δm₃ [kg]	0	0.1	0.2	0.3	0.4	0.5	0.6
k₁ [N/m]	2100
k₂ [N/m]	2100
k₃ [N/m]	2100
μ [1/s]	0.077466
κ [s]	0.00094753

Dynamic response of computational model at time instant t = 9_3s_

Iteration step	Displacements [mm]			Velocities [mm/s]			Accelerations [mm/s²]

	q₁	q₂	q₃	q˙1 {\dot q_1}	q˙2 {\dot q_2}	q˙3 {\dot q_3}	q¨1 {\ddot q_1}	q¨2 {\ddot q_2}	q¨3 {\ddot q_3}	z¨ \ddot z
1	−3.578	−7.647	−11.07	−21.71	−53.28	−83.45	144.3	378.9	603.6	−240.5
2	−3.044	−6.558	−9.494	−21.76	−53.65	−84.01	149.0	363.4	552.0	−240.5
3	−2.502	−5.469	−7.946	−22.22	−53.78	−83.11	151.1	344.6	503.3	−240.5
4	−1.936	−4.385	−6.465	−22.85	−53.46	−80.94	143.3	319.3	461.3	−240.5
5	−1.344	−3.319	−5.085	−23.17	−52.44	−77.80	122.1	287.1	427.9	−240.5
6	−0.742	−2.297	−3.826	−22.70	−50.48	−74.00	89.30	250.2	402.7	−240.5
7	−0.162	−1.346	−2.697	−21.10	−47.46	−69.80	51.76	212.2	383.6	−240.5

Dynamic response of computational model at time instant t = 5s_

Iteration step	Displacements [mm]			Velocities [mm/s]			Accelerations [mm/s²]

	q₁	q₂	q₃	q˙1 {\dot q_1}	q˙2 {\dot q_2}	q˙3 {\dot q_3}	q¨1 {\ddot q_1}	q¨2 {\ddot q_2}	q¨3 {\ddot q_3}	z¨ \ddot z
1	1.051	1.933	2.552	−4.227	−9.333	−13.72	−33.09	−53.37	−62.55	0
2	0.855	1.568	2.067	−3.574	−8.309	−12.53	−27.55	−43.05	−49.98	0
3	0.655	1.199	1.581	−3.028	−7.475	−11.56	−21.21	−32.47	−37.77	0
4	0.452	0.830	1.097	−2.602	−6.827	−10.80	−14.24	−21.83	−25.92	0
5	0.249	0.462	0.615	−2.305	−6.361	−10.23	−6.878	−11.30	−14.41	0
6	0.047	0.097	0.141	−2.140	−6.069	−9.842	0.541	−0.984	−3.222	0
7	−0.151	−0.261	−0.326	−2.102	−5.945	−9.627	7.686	9.010	7.682	0

Number of needed iterations – changes in momentum of system being identified_

Number of dynamic degrees of freedom d	Number of relations given by relation (15)	Number of unknowns n_u	Number of iterations n_r
2	2	10	5
3	3	19	7
4	4	31	8
5	5	46	10
6	6	64	11
7	7	85	13
8	8	109	14

DOI: https://doi.org/10.2478/sgem-2025-0008 | Journal eISSN: 2083-831X | Journal ISSN: 0137-6365

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Language: English

Page range: 89 - 102

Submitted on: Dec 23, 2024

Accepted on: Jan 27, 2025

Published on: Mar 25, 2025

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

numerical analyses,

input–output method,

change of system momentum,

identification of system parameters,

Related subjects:

Physics,

Mechanics and fluid dynamics

© 2025 Krzysztof Majcher, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

Previous article Volume 47 (2025): Issue 1 (March 2025)

Identification of mass, damping and stiffness matrices of multi degree of freedom system subjected to kinematic excitations

Figures & Tables

Figure 1:

Figure 2:

Figure 3:

Dynamic response of computational model at time instant t = 7_71s_

Computational model data in particular iteration steps_

Dynamic response of computational model at time instant t = 9_3s_

Dynamic response of computational model at time instant t = 5s_

Number of needed iterations – changes in momentum of system being identified_

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