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

Figure 1:

Figure 2:

Figure 3:

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 nu	Number of iterations nr
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