In this paper, the displacement response of a nonlinear stiffening two-degree-of-freedom system is investigated under a range of sinusoidal loadings with different frequencies and force amplitudes. The stiffness changes of this system had a quadratic relationship with the displacement value. Eight numerical methods including the 3rd-order acceleration method, the 2nd-order acceleration method, the Newmark method (linear acceleration method), the Newmark method (average acceleration method), the Wilson-Theta method, the central difference method, the Jennings method and the innovative method were used to calculate the displacement response of this system. Using the innovative method and using ∆t equal to 0.001 seconds, the displacement response of the system was calculated and was considered as the exact response of the system. Then, for ∆t s of 0.001, 0.002, 0.003, 0.004, 0.005, 0.006 and 0.007, using the 8 numerical methods introduced above, the displacement response of this system was calculated for PGAs of 1, 2, 3, 4 and 5 meters per second squared, and the values ​​of the root mean square error and the coefficient of variation of the error of these 8 methods were calculated for the ∆t s defined above, and the results of the work are presented in the form of bar graphs in the article. The results of the studies and observations showed that the error of the 3rd order acceleration method was not significantly different from the error of the 2nd order acceleration method for different ∆t s. Also, in both classes, the lowest error rate was generally related to the Wilson method and the highest was often related to the central difference method.
In this paper, a stiffening two-degree-of-freedom system was considered and a wide range of sinusoidal loadings were applied to it. Using 8 numerical methods, the displacement response of this system was calculated for different Δts and force amplitudes.
Overall, in the first class, Wilson's method had the lowest average error and the central difference method had the highest average error, and the 3rd-order acceleration and 2nd-order acceleration methods were ranked fifth and sixth, respectively.
In the second class, Wilson's method had the lowest average error and the central difference method had the highest average error, and the 3rd-order acceleration and 2nd-order acceleration methods were ranked fourth and sixth, respectively.
In terms of coefficient of variation, in the first class, the highest coefficient of variation was related to the central difference method and its lowest value was related to the Wilson method, and the 3rd-order acceleration and 2nd-order acceleration methods were ranked fifth and sixth, respectively.
In the second class, the highest coefficient of variation was related to the central difference method and its lowest value was related to the Wilson method, and the 2nd-order acceleration and 3rd-order acceleration methods were ranked fifth and sixth, respectively.