فهرست مطالب babak taqavi

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.

In this paper, the pseudo-acceleration spectra of the three accelerograms of El Centro, Naghan and Tabas, which were obtained by the linear interpolation method of stimulation (Jennings’s method) and also by using the cubic spline function, are compared with each other. Jennings' method is based on linear interpolation of excitation and is acceptable for short periods of time. In this research, three accelerograms of El Centro, Naghan and Tabas were considered, and their pseudo-acceleration response spectrum (base shear coefficient) was calculated for different damping values using linear interpolation method and interpolation with spline function. The considered damping were zero, two, five, ten and twenty percent. In order to determine the difference between the linear interpolation method and the spline function interpolation method, the time interval between the accelerometer points was divided into 2, 5, 10, 20, 50, 100, and 200 equal parts, respectively. Then, once with the linear interpolation method and again with the spline interpolation method, the internal points between the internal acceleration points were obtained and their pseudo-acceleration spectrum was calculated using the Jennings method and the interpolation method with the spline function. The calculation results showed that the pseudo-acceleration spectrum values calculated by the spline interpolation method in very short periods and low damping have a significant difference with the corresponding values of the pseudo-acceleration spectrum calculated by the linear interpolation method. It should be noted that the initial time interval between the accelerogram points of El Centro, Naghan and Tabas is 0.02 seconds.

Usually, by modeling the structures using the finite element method, their undamped free vibration frequencies are calculated analytically. In addition, the issue of accurate calculation of natural frequencies and the shape of vibration modes corresponding to them for bending systems that have distributed mass and elasticity and possibly a combination of several bending beams, sometimes requires solving complex mathematical equations and requires a relatively heavy mathematical work demands. Bending beams are beams whose axial deformations is insignificant compared to their bending deformations, and as a result, these members are assumed to be axially rigid. By using the conventional finite element method, the natural vibration frequencies of these beams can be obtained approximately. By increasing the number of finite elements used in the model, the calculation error of natural frequencies of vibration decreases. When the consistent-mass matrix is used, the frequency values obtained from the finite element method converge to the exact frequency values with larger values, while if the lumped-mass matrix is used, the frequency values obtained from the finite element method converge to the exact frequency values with smaller values. It should be noted that the consistent-mass matrix is non-diagonal, but the lumped-mass matrix is diagonal. The interpolation functions (shape functions) used for bending finite elements (beam elements) are polynomial functions of the 3rd degree. This bending finite element has two nodes, each node has one translational degree of freedom and one rotational degree of freedom. The new idea that came to the authors of this article is that instead of using polynomial functions, trigonometric and exponential interpolation functions are used to calculate the stiffness matrix and mass matrix of the finite element. In fact, these trigonometric and exponential functions are the solutions of the differential equation governing the free vibration of bending beams with distributed mass and elasticity. The argument of these trigonometric and exponential functions includes a parameter called beta, which is proportional to the square root of angular frequency of the bending beam. By changing this parameter in a suitable range and with a certain step, it is possible to plot the changes in the frequencies of the different modes of the studied prismatic beam in terms of beta. In this paper, three models were studied, which included a uniform cantilever beam, a uniform beam clamped at left side and simply supported at right side, and a uniform beam free at both ends. Using the conventional finite element method and using the consistent-mass matrix, these three models were analyzed and the approximate frequencies of the first few modes of these beams were calculated, which were greater than their corresponding exact values. In the innovative method presented in this article, a uniform beam was modeled with a finite element model with one translational degree of freedom and one rotational degree of freedom. The stiffness matrix and the mass matrix of this beam were calculated for different betas and having these two matrices, the first and second frequency values of this model were calculated for different beta values and its graph was drawn for different betas. The values of the maximum frequency of the first frequency are the same as the values of the minimum frequency of the second for certain betas, and by specifying these betas, the frequencies of different vibration modes can be accurately determined. The detected frequencies of different modes with this method had a very good match with their exact corresponding frequencies. For the second model investigated in this paper, one rotational degree of freedom was considered. Considering that this beam had only one rotational degree of freedom, therefore, by plotting the first frequency of this model for different betas and finding its minimum, the frequency values of different modes of this beam were obtained, which matched the exact values like the previous model very well. The third model was the same as the previous two models. The diagram of the first to fourth natural frequencies of this model was drawn for different betas. By having the approximate values of the frequencies of different modes obtained from the conventional finite element method and these diagrams, the frequencies of different modes of the model were identified, which were in good agreement with their corresponding exact values.