This paper investigates the displacement response of a nonlinear, stiffening single-degree-of-freedom system subjected to a range of sinusoidal loadings with varying frequencies and force amplitudes. The stiffness variation in this system follows a quadratic relationship with displacement. A total of nine numerical methods were utilized to compute the displacement response of the system. These methods include the third-order acceleration method, the second-order acceleration method, the Newmark’s linear acceleration method, the Newmark’s average acceleration method, the Wilson-Theta method, the central difference method, the Jennings method, the improved Jennings method, and a newly developed method. Using the developed method and a time step ∆t of 0.001 seconds, the system’s displacement response was computed and considered as the reference solution. Subsequently, for time steps of 0.001, 0.002, 0.004, 0.006, 0.008, and 0.01 seconds, the displacement response of the system was calculated using the aforementioned nine numerical methods under loading amplitudes of 200, 400, 600, 800, and 1000 Newtons. The root mean square error and the CV of the error for each method were calculated for the defined time steps. Results are presented as bar charts within the paper.
This paper investigates a nonlinear stiffening SDOF system without damping, with a mass of 1000 kg and an initial stiffness of 100,000 N/m. The mass and stiffness specifications of this system were taken from Chang (2009, pp. 289-297), where a two-degree-of-freedom shear system without damping is presented with a mass of 1000 kg for each floor.
In this study, the displacement response of the system under consideration, calculated with a time step of 0.001 seconds, is regarded as the accurate response of the system. To justify this, the studied nonlinear stiffening SDOF system was subjected to a range of sinusoidal loads with frequencies ranging from 1 Hz to 30 Hz, increasing by 0.1 Hz (291 frequencies), with a loading amplitude of 1000 Newtons (the maximum loading amplitude), and the system response was calculated with a maximum time step of 0.01 seconds. Then, using three methods (the innovative method, Newmark’s average acceleration method, and the Jennings method) and with a time step of 0.001 seconds, the system’s response was calculated and considered as the exact response. By reducing the number of points and changing the time interval from 0.001 seconds to 0.01 seconds, the maximum error and average errors of the nine methods discussed in this study were computed. Examining the results reveals that there is essentially no significant difference in the error values obtained by the three mentioned methods, especially for the average error values. However, since the innovative method assumes a constant stiffness value within each time step of 0.001 seconds, this method appears to offer a higher degree of accuracy.
Overall, the newly developed method, the improved Jennings method, and the original Jennings method, respectively, exhibited the lowest error, while the Wilson-Theta method showed the highest error. In terms of the coefficient of variation, Wilson-Theta, Newmark’s average acceleration, and the central difference method, respectively, had the lowest coefficients of variation in ascending order, with the improved Jennings method having the highest coefficient of variation