连续梁振动调整的快速解析

Study on a Fast Analytical Method of Adjustment of Continuous Beam Vibration

采用连续分段独立一体化积分法求解了连续梁自振角频率的解析表达式。首先采用弯曲-振动比拟法建立具有四阶导数的挠度微分方程,独立积分4次,得到挠度的通解。利用边界条件和连续性条件确定积分常数,得到挠度的解析表达式;然后根据最小能量原理得到了自振角频率的一次近似解析解;根据渐近法求解精确的振动微分方程得到更精确的挠度解析函数表达式,利用最小能量原理求得自振角频率的精确表达式。按照振动结构的同步失效准则和最优化准则对连续梁支座位置进行调整,得到了结构的固有角频率最优解的解析表达式。绘制了固有角频率随位置的变化曲线。工程实例表明,连续分段独立一体化积分法编程程式化,可以得到自振角频率最优的解析解。

A continuous subsection independently systematic integral method（ CSISIM） is used to solve the analytical expressions of the natural angular frequency of continuous beams vibration. First the forth-order differential deflection equations are derived by bending-vibration analogy method. Then the general solutions of beam deflection are obtained by independent forth-fold integration. Integral constants are determined by boundary conditions and continuity conditions to determine the analytical solution of deflection. According to the principle of minimum energy,we get the first order approximate analytical solution of the natural angular frequency. Then by progressive method,more accurate analytical solution is obtained by solving exact differential equations of the vibration. According to the principle of minimum energy,the exact expression of natural angular frequency can be solved. The support position of the continuous beam is adjusted by simultaneous failure criterion and optimization criterion of vibration structure. Analytic expression of the optimal solution of the nature angular frequency of the structure is obtained. The changing curve of the natural angular frequency versus position is plotted. The engineering example shows the CSISIM method is a general method suitable for computer stylized programming to solve. It can obtain the optimum analytical solution of natural angular frequency.