Abstract: 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.

Key words: rheonomic restraint, Euler-Bernoulli beam, steady response, differential algebraic equation, transient responses