| چکیده انگلیسی مقاله |
Given that the accelerograms recorded in various earthquakes are in the form of numerical data measured at short time intervals, the numerical evaluation of the dynamic response of structures is of great importance. In addition, during severe earthquakes, structures enter the nonlinear behavior range, and it is necessary to calculate the nonlinear behavior of structures using conventional numerical methods. Due to the small distance between the accelerogram points, the points are usually connected to each other by line segments. As a result, at the junction of the line segments, their slopes are not necessarily equal, and only the function values at their junctions are equal. Given that linear interpolation of excitation is one of the simplest interpolation methods, if the time intervals between the points are small compared to the natural period of the structure, linear interpolation is acceptable and provides sufficient accuracy. However, if the structure in question is highly rigid with a very small period–in other words, a very high frequency–the use of the linear excitation interpolation method can be challenging. There are various methods for interpolating a function over an interval, each with its own advantages and disadvantages. Approximating an arbitrary function over a closed interval can be very inaccurate due to the oscillatory nature of polynomials. Furthermore, a small change in the function over a subinterval can have a large effect on the interpolating polynomial. For this reason, in practice, it is preferable to divide the interval into small subintervals and reduce the degree of the interpolating polynomial as much as possible. This approach is called piecewise polynomial approximation. The simplest of these polynomials are linear polynomials (degree one). The graph of these functions is a broken line connecting the set of points (x0,y0), (x1,y1), ... and (xn,yn). One of the disadvantages of this method is the lack of differentiability at the ends of the subintervals. The geometric interpretation of this is the non-smoothness of the graph. To overcome this problem, in the cubic spline method, third-degree polynomial functions are used for each subinterval. By setting the values of these functions and their first and second derivatives equal at the connection points, equations are obtained for calculating the unknown coefficients of these polynomials. Of course, at the beginning and end points of the interval, the values of the function must also be substituted to obtain two other equations in terms of the unknown coefficients of the third-degree polynomials. Additionally, at the beginning and end points of the interval, free boundary conditions (setting the second derivative of the third-degree functions to zero at the beginning and end of the interval) or restricted boundary conditions (setting the first derivative of the third- degree functions to zero at the beginning and end of the interval) must be used to obtain two other equations in terms of the unknown coefficients of the third- degree polynomials. By considering all these equations, the unknown coefficients of the third-degree polynomials are calculated. It should be noted that free boundary conditions were used in this study. In this article, three accelerograms–El Centro, Tabas and Naghan–were considered and their velocity and displacement records were obtained by numerical integration. The time interval between the points of these three accelerograms was 0.02 seconds. Then, the number of points of records in the acceleration, velocity and displacement records was reduced with reduction rates of 1, 2, 3, 4 and 5. In the next step, the deleted data were calculated using linear interpolation and cubic spline interpolation, and the root mean square error (RMSE) of the calculated values was computed by comparing them to their actual values for both interpolation methods. Subsequently, 21 records of acceleration, velocity and displacement from different earthquakes were downloaded from the PEER website. Unlike the previous three records, these records had varying time steps. Similar to the previous three records, the RMSE was calculated for data reduction rates of 1, 2, 3, 4 and 5 using both linear interpolation and spline interpolation. The comparison of these errors showed that the linear interpolation error for displacement records is higher than the spline interpolation error for the corresponding velocity records. Additionally, it was observed that the linear interpolation error for velocity records is higher than the spline interpolation error for the corresponding acceleration records. Moreover, with the increase in data reduction rate, the ratio of linear interpolation error to the corresponding error of spline interpolation had a decreasing trend. Finally, power relationships were developed for the average ratio of the linear interpolation error to the spline interpolation error as a function of the data reduction rate. These relationships showed a strong correlation with their actual corresponding values. |
| کلیدواژههای انگلیسی مقاله |
درونیابی اسپلاین مکعبی,درونیابی خطی,جذر میانگین مربعات خطا,رکوردهای شتاب,سرعت و تغییر مکان زلزله,نوفه,گام زمانی,برازش منحنی |