Gan Jiarong, Yuan Hong, Li Shanqing, Peng Qifeng, Zhang Huanliang
MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, China.
Heliyon. 2021 Jan 6;7(1):e05876. doi: 10.1016/j.heliyon.2020.e05876. eCollection 2021 Jan.
The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of R-function, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method.
双参数地基上扁球壳的自由振动控制微分方程是一个四阶微分方程。利用中间变量，将该四阶微分方程降为两个低阶微分方程。第一个低阶微分方程是亥姆霍兹方程。本文提出了一种二维亥姆霍兹算子的新方法，其中需要对亥姆霍兹方程中包含的贝塞尔函数进行适当处理以消除奇异性。利用本文提出的方法将第一个低阶微分方程转化为积分方程。第二个低阶微分方程是拉普拉斯方程，利用现有方法将其转化为积分方程。然后根据中矩形公式对两个积分方程进行离散化，并通过MATLAB编程得到相应的解。本文利用R函数理论选择合适的边界方程以消除奇异性。基于R函数的性质，亥姆霍兹方程和拉普拉斯方程的组合方法可以解决双参数地基上不规则扁球壳的自由振动问题。给出了五个例子来验证该方法的可行性。
