Fig. 1.
![The two substructures : soil and rigid structure with soil-structure interface [38]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/67f3415f8d1bec042eac83d9/j_ama-2025-0006_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKJ3UZ6ZQE%2F20251216%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20251216T003129Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEI3%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaDGV1LWNlbnRyYWwtMSJHMEUCIQDpyY6BGds9G9qHXSP7em2iaCW2MXj7tB8Sa1xqPWUhLAIgK26R4Q7SRT45HqB4BMWjdeETlLptnk20BDA1B%2BomLkwqvAUIVhACGgw5NjMxMzQyODk5NDAiDMQcAPfQbbnvcaRdTCqZBfTpdRSlEE2VrVVoTkhHNnb8TQnfWpediPLGqxssbUO7p0ofMmojFP7kL0L4AFrdzA2NwMlerEzo%2BR%2B9xlTnJBBKo%2BbMyGPyoMqBocz%2Bs2aGSj9rShesO9RiMg2veifXVg4CddQrgHSdss5jhHeYp02VvNNTofAfUM%2F4hkJx3mVnMiBHMcboi9KXP0l94s4%2BCm9dfreVKfPpHRNWnBUj4RyVmePIGEiirqC%2FlWUfMvQclj4e6tSfuomUE40REgbatc0FzG0k57pw0nAEkOLBFlDnwduvmBRzoN4J5qfhbg%2FdM0Af9u1n8ZkmB%2BbQBxQFPhZ9B2nhgxbKVBgpN8NS3s2fT5LYkMUK2UFR7Bm65BudBjkoGgjlF8GWGcPKoE6dtX2kv5ju7z3QvWUyNhYQggVbsKktVnl%2BgU5H0KP1yadUqqO13adIzZO6%2Fdf3Xucfmb2tJDt9e5pq%2BeXNcPSa4%2BYvGMwHDP4d2F7aqM0zx1fqeC1AyQ8WS3boEKkHTYRJqM3VnFPaKZH0rov%2BuC128GHysHyloeTloTjxM2oLDWZFIne2bjYkdTgVEuiyABHWu2xN8eUg40KWBooGsCrYzOZteMeurJFvXAolklfSpmpr37AXWfWA0M%2FbZrjlSkEdsPWSLnKucIGefYnanAVMJfznAapByHeOvGgO%2BAGs3XKd%2FnROm7AiUyjBgg28TIbpJnQcRyTUDAQo5oX5oEGmuHr0bdM%2FWlA%2B7RIxLmmiAo%2BgY0FDGBr4T%2BubA4uTrOSWJ125654R4EVheUPDDDDTzjFJjc7NrxkIOPwZUBU%2BHTAYdZWnIWK6G8JeMP22JXIFarUbLau7fG97xcgxZZ2WmftALHulH9%2Fw%2BEDcZvGLPW4xqhXUvlQ6S0slMP3kgcoGOrEB6WpLv7dsu7WR6FhJA8yon5dkF182gH%2BK3yK2sv0%2BDnBPsCAyBh1%2FmAsmT4Lo8KBuUkXTBHBNoTWSsrN5yh199uvYmWhOTxzxMS4QIuB7bGT0XvH6Tmb9KmvrRIuUgjSewYlzK9re1OC7tK6cV6oyoi4tCEeseOQ9lOHejlQrxFUejdxLnSwzBPf%2F0O475gnZkPvcNYDkTciaVs2HzdvXZtNigiW%2B9%2BX06SN76fspwBlH&X-Amz-Signature=69ff47107267d19d133280e6fc459d6a5bfbe7a9937f961df141b7afa5496fe3&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)
Fig. 2.

Fig. 3.

Fig. 4.
![Flowchart of solving the state equation with active control [18]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/67f3415f8d1bec042eac83d9/j_ama-2025-0006_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKJ3UZ6ZQE%2F20251216%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20251216T003129Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEI3%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaDGV1LWNlbnRyYWwtMSJHMEUCIQDpyY6BGds9G9qHXSP7em2iaCW2MXj7tB8Sa1xqPWUhLAIgK26R4Q7SRT45HqB4BMWjdeETlLptnk20BDA1B%2BomLkwqvAUIVhACGgw5NjMxMzQyODk5NDAiDMQcAPfQbbnvcaRdTCqZBfTpdRSlEE2VrVVoTkhHNnb8TQnfWpediPLGqxssbUO7p0ofMmojFP7kL0L4AFrdzA2NwMlerEzo%2BR%2B9xlTnJBBKo%2BbMyGPyoMqBocz%2Bs2aGSj9rShesO9RiMg2veifXVg4CddQrgHSdss5jhHeYp02VvNNTofAfUM%2F4hkJx3mVnMiBHMcboi9KXP0l94s4%2BCm9dfreVKfPpHRNWnBUj4RyVmePIGEiirqC%2FlWUfMvQclj4e6tSfuomUE40REgbatc0FzG0k57pw0nAEkOLBFlDnwduvmBRzoN4J5qfhbg%2FdM0Af9u1n8ZkmB%2BbQBxQFPhZ9B2nhgxbKVBgpN8NS3s2fT5LYkMUK2UFR7Bm65BudBjkoGgjlF8GWGcPKoE6dtX2kv5ju7z3QvWUyNhYQggVbsKktVnl%2BgU5H0KP1yadUqqO13adIzZO6%2Fdf3Xucfmb2tJDt9e5pq%2BeXNcPSa4%2BYvGMwHDP4d2F7aqM0zx1fqeC1AyQ8WS3boEKkHTYRJqM3VnFPaKZH0rov%2BuC128GHysHyloeTloTjxM2oLDWZFIne2bjYkdTgVEuiyABHWu2xN8eUg40KWBooGsCrYzOZteMeurJFvXAolklfSpmpr37AXWfWA0M%2FbZrjlSkEdsPWSLnKucIGefYnanAVMJfznAapByHeOvGgO%2BAGs3XKd%2FnROm7AiUyjBgg28TIbpJnQcRyTUDAQo5oX5oEGmuHr0bdM%2FWlA%2B7RIxLmmiAo%2BgY0FDGBr4T%2BubA4uTrOSWJ125654R4EVheUPDDDDTzjFJjc7NrxkIOPwZUBU%2BHTAYdZWnIWK6G8JeMP22JXIFarUbLau7fG97xcgxZZ2WmftALHulH9%2Fw%2BEDcZvGLPW4xqhXUvlQ6S0slMP3kgcoGOrEB6WpLv7dsu7WR6FhJA8yon5dkF182gH%2BK3yK2sv0%2BDnBPsCAyBh1%2FmAsmT4Lo8KBuUkXTBHBNoTWSsrN5yh199uvYmWhOTxzxMS4QIuB7bGT0XvH6Tmb9KmvrRIuUgjSewYlzK9re1OC7tK6cV6oyoi4tCEeseOQ9lOHejlQrxFUejdxLnSwzBPf%2F0O475gnZkPvcNYDkTciaVs2HzdvXZtNigiW%2B9%2BX06SN76fspwBlH&X-Amz-Signature=1511194556e0bba8a442bdbe295457e67b3f28b80ecd062398b46be08293b46e&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)
Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

The natural frequencies of the studied model with and without SSI (rad/s)
| Floor | SSI | Fixed |
|---|---|---|
| 1 | 14,514 | 17,236 |
| 2 | 43,162 | 48,295 |
| 3 | 68,091 | 69,789 |
![The two substructures : soil and rigid structure with soil-structure interface [38]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/67f3415f8d1bec042eac83d9/j_ama-2025-0006_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKJ3UZ6ZQE%2F20251216%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20251216T003129Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEI3%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaDGV1LWNlbnRyYWwtMSJHMEUCIQDpyY6BGds9G9qHXSP7em2iaCW2MXj7tB8Sa1xqPWUhLAIgK26R4Q7SRT45HqB4BMWjdeETlLptnk20BDA1B%2BomLkwqvAUIVhACGgw5NjMxMzQyODk5NDAiDMQcAPfQbbnvcaRdTCqZBfTpdRSlEE2VrVVoTkhHNnb8TQnfWpediPLGqxssbUO7p0ofMmojFP7kL0L4AFrdzA2NwMlerEzo%2BR%2B9xlTnJBBKo%2BbMyGPyoMqBocz%2Bs2aGSj9rShesO9RiMg2veifXVg4CddQrgHSdss5jhHeYp02VvNNTofAfUM%2F4hkJx3mVnMiBHMcboi9KXP0l94s4%2BCm9dfreVKfPpHRNWnBUj4RyVmePIGEiirqC%2FlWUfMvQclj4e6tSfuomUE40REgbatc0FzG0k57pw0nAEkOLBFlDnwduvmBRzoN4J5qfhbg%2FdM0Af9u1n8ZkmB%2BbQBxQFPhZ9B2nhgxbKVBgpN8NS3s2fT5LYkMUK2UFR7Bm65BudBjkoGgjlF8GWGcPKoE6dtX2kv5ju7z3QvWUyNhYQggVbsKktVnl%2BgU5H0KP1yadUqqO13adIzZO6%2Fdf3Xucfmb2tJDt9e5pq%2BeXNcPSa4%2BYvGMwHDP4d2F7aqM0zx1fqeC1AyQ8WS3boEKkHTYRJqM3VnFPaKZH0rov%2BuC128GHysHyloeTloTjxM2oLDWZFIne2bjYkdTgVEuiyABHWu2xN8eUg40KWBooGsCrYzOZteMeurJFvXAolklfSpmpr37AXWfWA0M%2FbZrjlSkEdsPWSLnKucIGefYnanAVMJfznAapByHeOvGgO%2BAGs3XKd%2FnROm7AiUyjBgg28TIbpJnQcRyTUDAQo5oX5oEGmuHr0bdM%2FWlA%2B7RIxLmmiAo%2BgY0FDGBr4T%2BubA4uTrOSWJ125654R4EVheUPDDDDTzjFJjc7NrxkIOPwZUBU%2BHTAYdZWnIWK6G8JeMP22JXIFarUbLau7fG97xcgxZZ2WmftALHulH9%2Fw%2BEDcZvGLPW4xqhXUvlQ6S0slMP3kgcoGOrEB6WpLv7dsu7WR6FhJA8yon5dkF182gH%2BK3yK2sv0%2BDnBPsCAyBh1%2FmAsmT4Lo8KBuUkXTBHBNoTWSsrN5yh199uvYmWhOTxzxMS4QIuB7bGT0XvH6Tmb9KmvrRIuUgjSewYlzK9re1OC7tK6cV6oyoi4tCEeseOQ9lOHejlQrxFUejdxLnSwzBPf%2F0O475gnZkPvcNYDkTciaVs2HzdvXZtNigiW%2B9%2BX06SN76fspwBlH&X-Amz-Signature=69ff47107267d19d133280e6fc459d6a5bfbe7a9937f961df141b7afa5496fe3&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)


![Flowchart of solving the state equation with active control [18]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/67f3415f8d1bec042eac83d9/j_ama-2025-0006_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKJ3UZ6ZQE%2F20251216%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20251216T003129Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEI3%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaDGV1LWNlbnRyYWwtMSJHMEUCIQDpyY6BGds9G9qHXSP7em2iaCW2MXj7tB8Sa1xqPWUhLAIgK26R4Q7SRT45HqB4BMWjdeETlLptnk20BDA1B%2BomLkwqvAUIVhACGgw5NjMxMzQyODk5NDAiDMQcAPfQbbnvcaRdTCqZBfTpdRSlEE2VrVVoTkhHNnb8TQnfWpediPLGqxssbUO7p0ofMmojFP7kL0L4AFrdzA2NwMlerEzo%2BR%2B9xlTnJBBKo%2BbMyGPyoMqBocz%2Bs2aGSj9rShesO9RiMg2veifXVg4CddQrgHSdss5jhHeYp02VvNNTofAfUM%2F4hkJx3mVnMiBHMcboi9KXP0l94s4%2BCm9dfreVKfPpHRNWnBUj4RyVmePIGEiirqC%2FlWUfMvQclj4e6tSfuomUE40REgbatc0FzG0k57pw0nAEkOLBFlDnwduvmBRzoN4J5qfhbg%2FdM0Af9u1n8ZkmB%2BbQBxQFPhZ9B2nhgxbKVBgpN8NS3s2fT5LYkMUK2UFR7Bm65BudBjkoGgjlF8GWGcPKoE6dtX2kv5ju7z3QvWUyNhYQggVbsKktVnl%2BgU5H0KP1yadUqqO13adIzZO6%2Fdf3Xucfmb2tJDt9e5pq%2BeXNcPSa4%2BYvGMwHDP4d2F7aqM0zx1fqeC1AyQ8WS3boEKkHTYRJqM3VnFPaKZH0rov%2BuC128GHysHyloeTloTjxM2oLDWZFIne2bjYkdTgVEuiyABHWu2xN8eUg40KWBooGsCrYzOZteMeurJFvXAolklfSpmpr37AXWfWA0M%2FbZrjlSkEdsPWSLnKucIGefYnanAVMJfznAapByHeOvGgO%2BAGs3XKd%2FnROm7AiUyjBgg28TIbpJnQcRyTUDAQo5oX5oEGmuHr0bdM%2FWlA%2B7RIxLmmiAo%2BgY0FDGBr4T%2BubA4uTrOSWJ125654R4EVheUPDDDDTzjFJjc7NrxkIOPwZUBU%2BHTAYdZWnIWK6G8JeMP22JXIFarUbLau7fG97xcgxZZ2WmftALHulH9%2Fw%2BEDcZvGLPW4xqhXUvlQ6S0slMP3kgcoGOrEB6WpLv7dsu7WR6FhJA8yon5dkF182gH%2BK3yK2sv0%2BDnBPsCAyBh1%2FmAsmT4Lo8KBuUkXTBHBNoTWSsrN5yh199uvYmWhOTxzxMS4QIuB7bGT0XvH6Tmb9KmvrRIuUgjSewYlzK9re1OC7tK6cV6oyoi4tCEeseOQ9lOHejlQrxFUejdxLnSwzBPf%2F0O475gnZkPvcNYDkTciaVs2HzdvXZtNigiW%2B9%2BX06SN76fspwBlH&X-Amz-Signature=1511194556e0bba8a442bdbe295457e67b3f28b80ecd062398b46be08293b46e&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)






| Floor | SSI | Fixed |
|---|---|---|
| 1 | 14,514 | 17,236 |
| 2 | 43,162 | 48,295 |
| 3 | 68,091 | 69,789 |
© 2025 Reteri Khadidja, Megnounif Abdellatif, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.