受多点横向非定常约束梁的稳态与瞬态响应

研究受有多点横向非定常约束梁的动态响应问题。利用横向非定常约束为周期函数梁的微分代数方程的线性特点，确定出多基频周期性非定常约束激励下梁稳态响应的解析解；利用截断模态的微分代数方程确定出受有横向非定常约束梁的瞬态响应。为此，首先利用无阻尼截断模态微分代数方程的齐次形式，将欧拉-伯努利梁的等效多跨梁模态表示为简单边界条件下模态函数的线性组合，再利用得到的模态函数与位移影响函数重新将微分代数方程表示为常微分方程的形式，进而得到横向非定常约束作用下梁的瞬态响应解的积分形式。在此过程中，研究了微分代数方程齐次式的特征值及其数量、求解方法等相关问题。通过单点多频率的横向非定常约束作用的梁及多点单频横向非定常约束作用梁的算例，重点分析了非定常约束位置与基频对梁稳态响应的影响。结果表明：在欠阻尼下，两种梁模态响应的极大值均在等效多跨梁的各个主频附近；单点横向非定常约束作用响应极小值在简单边界条件梁的各个主频附近，而多点横向非定常约束作用梁响应极小值的分布比较复杂。算例也说明了所提出的方法是正确、有效的。

On the Steady and Transient Responses of the Beam with Many Transverse Rheonomic Restraints

The steady and transient responses of the beam with more than one transverse rheonomic restraints are studied. By virtue of linear characteristic of the differential algebraic equations under the periodical transverse rheonomic restraints, the analytical solution for steady responses of the beam is derived. And for transient responses, truncated modes of the differential algebraic equations are utilized. For this purpose, based on the homogeneous differential algebraic equations without damping, the mode of the equivalent multi span beam is expressed as a linear combination of modal functions of a Euler-Bernoulli beam with simple boundary conditions, and further, an integral form, a vital part in determining the transient responses, is obtained. The eigenvalues of homogeneous differential algebraic equations and its total number, the method to find them are addressed. Finally, two examples, one beam with a rheonomic restraint but multi frequencies, and the other one with many rheonomic restraints but sole frequency, are presented to show the impact of location and frequencies of rheonomic restraint on the responses of the beams. It is shown that the maximum values of the mode response, in both cases, occur near each natural frequency of the equivalent multi span beam. But the minimum values of the mode response with a rheonomic restraint occur near each natural frequency, whereas distribution of minimum values for that with many rheonomic restraints is much more complicated. The two examples also demonstrate that the proposed approaches are useful and effective.