变截面杆轴向和扭转振动分析的摄动法

Perturbation method for longitudinal and torsional vibration analysis of bar with variable cross-section

基于变截面杆是工程结构中应用广泛的承力构件,采用摄动法研究变截面杆轴向和扭转固有振动。推导求解二阶线性齐次微分方程的摄动法,并采用该方法求出变截面杆轴向和扭转振动平衡微分方程的近似解析解;结合边界条件得到固有频率的特征方程,计算变截面杆轴向和扭转振动的固有频率。通过算例分析变截面杆轴向和扭转振动固有频率与形状参变量之间的变化规律,确定变截面杆形状参变量变化时固有频率极大值和极小值。研究结果表明:该方法简便实用,且有良好的近似性。

Considering Variable cross-section bar is an important structural member in engineering structure, the longitudinal and torsional vibration of bar with variable cross-section was studied by perturbation method. Firstly, the perturbation method which is of particular importance in solving linear homogeneous differential equation of second order with variable coefficients was derived. Then the approximate analytical solution of the differential equation of longitudinal and torsional vibration of bar with variable cross-section was constructed using perturbation method, and the characteristic equations of natural frequencies were obtained considering the boundary constraints. Finally, as the numerical examples, the natural frequencies of longitudinal and torsional vibration of bar with variable cross-section were calculated. In the calculation example the variation law of natural frequencies with shape parametric variables was presented, and the maximum and minimum of the natural frequencies were given. The results show that the method is simple and practicable with high precision.