In the dynamics of linear structures, the impulse response function is of fundamental interest. In some cases one examines the short term response wherein the disturbance is still local and the boundaries have not yet come into play, and for such short-time analysis the geometrical extent of the structure may be taken as unbounded. Here we examine the response of slender beams to angular impulses. The Euler-Bernoulli model, which does not include rotary inertia of cross sections, predicts an unphysical and unbounded initial rotation at the point of application. A finite length Euler-Bernoulli beam, when modelled using finite elements, predicts a mesh-dependent response that shows fast large-amplitude oscillations setting in very quickly. The simplest introduction of rotary inertia yields the Rayleigh beam model, which has more reasonable behaviour including a finite wave speed at all frequencies. If a Rayleigh beam is given an impulsive moment at a location away from its boundaries, then the predicted behaviour has an instantaneous finite jump in local slope or rotation, followed by smooth evolution of the slope for a finite time interval until reflections arrive from the boundary, causing subsequent slope discontinuities in time. We present a detailed study of the angular impulse response of a simply supported Rayleigh beam, starting with dimensional analysis, followed by modal expansion including all natural frequencies, culminating with an asymptotic formula for the short-time response. The asymptotic formula is obtained by breaking the series solution into two parts to be treated independently term by term, and leads to a polynomial in time. The polynomial matches the response from refined finite element (FE) simulations.

翻译：在线性结构动力学中，脉冲响应函数具有基础性意义。在某些情况下，我们研究扰动仍局限于局部且边界尚未产生影响的短时响应，此时结构的几何范围可视为无界。本文研究了细长梁对角脉冲的响应。未包含截面转动惯量的欧拉-伯努利模型预测了加载点处非物理且无界的初始转角。采用有限元建模的有限长欧拉-伯努利梁，会呈现网格依赖的响应特征，即快速出现大幅振荡。引入转动惯量的最简单模型即为瑞利梁模型，其行为更合理，所有频率均具有有限波速。若对远离边界的瑞利梁施加脉冲弯矩，其预测行为表现为局部斜率或转角发生瞬时有限跳跃，随后在有限时间区间内斜率平滑演化，直至边界反射到达导致后续时间上的斜率间断。本文对简支瑞利梁的角脉冲响应进行了详细研究，从量纲分析出发，接着开展了包含所有固有频率的模态展开，最终推导出短时响应的渐近公式。该渐近公式通过将级数解分解为两个独立逐项处理的部分得到，并形成时间多项式。此多项式与精细化有限元仿真结果吻合。