Improvement of the Linear Interpolation Method of Excitation Using the Spline Interpolation Function to Numerically Calculate the Response Coefficient of Linear One-Degree-Of-Freedom Systems

Document Type : Research Note

Authors

1 Assistant Professor, Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran

2 Ph.D. Candidate, Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran

3 Ph.D. Candidate, Department of Civil Engineering, Noshirvani University of Technology, Babol, Iran

4 M.Sc. Student, Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran

Abstract

In this article, the Jennings method has been improved by using the third-order spline interpolation function. In this research, in order to be able to compare the advantages and disadvantages of the Jennings method, which is based on exact relationships and the assumption of linear changes of the excitation, compared to the spline interpolation method, a damped one-degree-of-freedom linear system under sinusoidal harmonic loading is considered has been taken. Exact values of the deformation response coefficient of this system are available for different excitation frequencies. The approximate values of the deformation response coefficient of this system for different excitation frequencies were calculated assuming linear interpolation of excitation and also assuming interpolation with spline function and were compared with their exact corresponding values. This work was done for two, five, ten and twenty percent damping. The results of the work indicated that when the number of points by which the sine wave is approximated is small and also the amount of damping is low, the interpolation with the spline function has a significantly higher accuracy than the linear interpolation mode. Another noteworthy point is the high sensitivity of the cubic spline interpolation to the level of system damping, so that for a certain number of divisions, the error value is highly sensitive to the damping value. Thus, with the increase of damping, the sign of the error changes and its value increases strongly for the number of large divisions.  This phenomenon is not seen at all in linear interpolation.
Therefore, considering that in interpolation using the spline method, the continuity of the slope and the second derivative are maintained in the internal points, it is suggested to use this method to calculate the dynamic response of linear systems and the results obtained with the results of Jennings' method should be compared and appropriate decisions should be made if there is a significant difference in the analysis results obtained from these two methods.
It is important to mention that the execution time of the computer program related to the cubic spline interpolation method is longer than the Jennings method. Because in the spline interpolation method, it is necessary to solve the system of linear equations with the number of unknowns equal to four times the number of intervals, while in the excitation linear interpolation method, it is not necessary to solve the system of equations at all and simply by having the velocity and displacement at the beginning of the step and assuming that the excitation is linear during the desired time step, the coefficients A, B, C, D, A', B', C', D' were calculated and using this eight coefficient, velocity and displacement at the end of the time step are obtained. It should be noted that if the time interval between different accelerogram points is constant, it is sufficient that the coefficients A, B, C, D, A', B', C', D' are calculated only once.

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History

  • Receive Date: 11 July 2022
  • Revise Date: 12 March 2023
  • Accept Date: 13 May 2023

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