Numerical Solutions for Nonlinear Vibrations of the Mathematical Pendulum Based on a Modified Frequency Formulation: Location Point Configurations

A	T_RK	T_FLP Err(%)	T_kLP Err(%)	T_MLP Err(%)	T_HPM1 Err(%)	T_HPM2 Err(%)	T_MHPM Err(%)	T_HBM Err(%)
0.1	6.2871	6.2871	6.2871	6.2871	6.2871	6.2871	6.2871	6.2871
0.1	6.2871	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
0.3	6.3187	6.3187	6.3187	6.3187	6.3187	6.3187	6.3187	6.3187
0.3	6.3187	0.0001	0.0003	0.0002	0.0003	0.0000	0.0000	0.0003
0.5	6.3828	6.3827	6.3826	6.3829	6.3826	6.3828	6.3828	6.3827
0.5	6.3828	0.0004	0.0024	0.0012	0.0022	0.0000	0.0000	0.0022
1.0	6.7000	6.6994	6.6975	6.7014	6.6961	6.6999	6.6999	6.6971
1.0	6.7000	0.0085	0.0362	0.0206	0.0581	0.0018	0.0015	0.0429
1.4	7.1523	7.1493	7.1410	7.1583	7.1222	7.1504	7.1505	7.1372
1.4	7.1523	0.0418	0.1588	0.0833	0.4202	0.0267	0.0249	0.2110
1.8	7.8582	7.8456	7.8173	7.8773	7.6987	7.8419	7.8424	7.7963
1.8	7.8582	0.1592	0.5206	0.2441	2.0296	0.2075	0.2001	0.7869
2.0	8.3498	8.3249	8.2758	8.3820	8.0238	8.3083	8.3094	8.2303
2.0	8.3498	0.2978	0.8853	0.3864	3.9041	0.4963	0.4832	1.4304
2.5	10.3232	10.1737	9.9878	10.4237	8.8029	9.9763	9.9805	9.7238
2.5	10.3232	1.4480	3.2487	0.9735	14.726	3.3597	3.3193	5.8064
3.0	16.1555	14.2775	13.4646	15.7941	9.2483	12.7146	12.7271	12.0333
3.0	16.1555	11.6246	16.6565	2.2371	42.7546	21.2986	21.2217	25.5159

Language: English

Page range: 209 - 218

Submitted on: Aug 29, 2025

Accepted on: Jul 3, 2026

Published on: Aug 5, 2026

Published by: Slovak Academy of Sciences, Institute of Measurement Science

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

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© 2026 Jingying Gao, Qingmei Bai, Jiahui Yu, Xiaopeng Li, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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