Analysis of natural frequency of beams with power-function-type continuous variable section
To provide a design reference for the flexural vibration of variable section amplitude rods,in this paper,the natural frequencies of beams with variable section are obtained by using analytical solution method for differential equation.First,a model is established for beam with variable cross-section.The analytical form of the deflection of the beam is derived,and the bending vibration frequency equation is obtained under different boundary conditions.For the cross section of a rod,when the power exponent of the function for moment of inertia is 4 larger than that of the function for area,the exact form of the natural frequency equation is obtained.The approximate form of the natural frequency equation is obtained for the case of the difference of the power exponent being not equal to 4.Then for four kinds of cross-section,the frequencies of the beams are obtained for different boundary conditions.It is found that compared with the results by the Rayleigh-Ritz method the relative derivative is less than 5%,and this proves the accuracy of the presented analytical method.The presented method has the advantage of being a fast-solving method,which can also be used to analyze the influence of the geometric parameters on the natural frequencies.Finally,it can be concluded that the dimensionless natural frequencies of variable section beams increase with the index of the varying cross section if the boundary and other parameters keep unchanged.With the increasing of the shape coefficient,the dimensionless natural frequency of the same order of the beam gradually decreases except for the first order natural frequency of the clamped-free beam.
Beams with variable sectionNatural frequencyAnalytical method for the differential equation