【摘要】 本文首先用分析方法求出了圆錐壳的振型函数和横向振动固有频率的精确解,然后为了实际应用的目的,建議了一种簡化計算方法。 文内采用了扁壳理論形式或称Donnell形式的运动微分方程組;在忽略切向慣性力分量的假设下导出了以一个横向位移函数表示的独立方程,从而得到了振型函数的冪級数解答,由此可看出圆錐壳的振型函数具有非周期性和幅度递增很快的振蕩特性。 鉴于上述运算过于繁重,故再提出一种簡化計算方案,其物理概念为:給壳体上任一元素建立一物理模型如图2所示,将振动时产生的薄膜张力和抗弯（扭）力矩的弹性恢复作用分別看作二个弹簧k₁和k₂,壳体元素即图中质量m,这样就得到一个并联弹簧的单自由度系統;于是整个壳体就相当于由无限多个这种单自由度系統組成,而弹簧刚度則是坐标的函数。由于該模型的固有频率可由迭加关系——ω²=k₁/m+k₂/m=ω₁²+ω₂²計算,故可推出,錐壳振动可以按无矩理論和純力矩理論分别計算出ω₁和ω₂,然后迭加而得固有頻率ω。这样将使計算大为簡化,且能得到滿意結果。文末提供了实驗驗証数据。 本文的計算可用于各种錐度的圆錐壳。同时为了論証上述方案的正确性,計算了a=0即圆柱壳情况;并給出了计算圆柱壳振动的簡便方法。为了计算完整的圆錐壳,还採討了錐尖处边界条更多还原

【Abstract】 This paper contains two parts: an exact solution of vibration modes and transverse cigenfrequencies of conical shells is studied first, and then is presented a simplified method or computation for practical uses.Donnell type differential equations with variable coefficients are used. Under the assumption of neglecting the tangential components (tangential to median surface) of inertia force, and taking the solution as the following type:an uncoupled equation for solving functions of transverse vibration modes is deduced after using some differentia! operators, as the following:Other two equations, showing the relations between U(x) and W(x), and V(x) and W (x) respectively areliquation (2; gives power-series solutions for W(x), liaving a unperiodic osoillating character and rapidly increasing amplitude, so are the solutions of U(x) and V(x). Unfortunately, the convergence of this series solution is poor, so that limits its practical applications.In order to reduce the computing works, a simplified method is recommended. The physical conception of this method is illustrated as following: Cutting one element from the shell arbitrarily, we shall use a physical model to represent this element (see fig- 2) When vibrating, the elastic restoring effects due to membrane extensional forces and bending (and twisting) moments are looked as two springs in parallel as fig. 2, while the element is the mass m. Then this element of shell is equivalent to the parallel springs system with single degree of freedom, and the whole shell can be represented by the sum of infinite systems of this single type, where the stiffness of springs are functions of the co-ordinates. It is known, that the eigenfrequency of the parallel springs system isThis relation shows us to calculate the eigenfrequencies of conical and cylindrical shells(and other shells) by superposing the results which are obtained from membrane theoryand pure bending theory respectively, i.e., the result w1 is solved by using the following equation :together with cqu. (3) and (4), while the result w2 is obtained by solving the equation:Ihe answer can then be calculated, according the relation w2 = w12 + w22.The analytical method presented in this paper can be used for shells with arbitrary conical angle a and various boundary conditions (the conditions at the vertex point of complete conical shell is discussed).更多还原