欧拉-伯努利直梁弯曲振动内力解的一种新形式

针对欧拉-伯努利直梁受简谐载荷作用下的弯曲振动问题，在得到响应后再次采用达朗贝尔原理，给出了一种新形式的弯曲振动内力解，该解直观显示了简谐力和惯性力对弯曲振动内力的贡献。基于傅里叶级数严格证明了其与教材中利用梁挠曲线近似微分方程和弯曲内力微分关系求出的内力完全等价，为加深理解及应用傅里叶级数、静力学理论、达朗贝尔原理和梁弯曲理论提供了一个综合性较高的案例，期望能助力提升基础力学教学效果。

A NEW FORM OF THE BENDING VIBRATION INTERNAL FORCES FOR EULER-BERNOULLI STRAIGHT BEAMS

The article focuses on the bending vibration problem of Euler Bernoulli straight beams under the harmonic load. A new form of bending internal force is proposed by using d'Alembert principle again after obtaining the response. This solution directly shows the contributions of the harmonic load and the inertial force to the bending internal forces, respectively. It is proved that it is equivalent to the bending internal force obtained by using the differential equation for deflection of beams and the differential relationship between the beam internal force in the textbook based on Fourier series. This provides a comprehensive case for deeply understanding and application of Fourier series, statics theory, d'Alembert principle and beam bending theory. It is expected to improve the teaching effectiveness of basic mechanics.