Typical experimental prototypes of flexible robot arms are constituted as a rotating beam with distributed flexibility. The simplest example of flexible manipulators is a flexible slewing beam constrained to move in a horizontal plane. The beam is usually actuated by a motor and may carry a payload. It occurs in such diverse applications as a link of a flexible robot arm, a flexible antenna, a helicopter blade, and also as part of a flexible structure [1][2][3][4][5][6].
The problem of bending vibration of a rotating beam undergoing large motions has been considered for many decades. For example, a work by Lo [7] considered the non-linear term which arises from the Coriolis acceleration, while reference [8] included all inertia effects, geometrical non-linearities, and coupling of extensional-flexural deformations. Eick and Mignolet [9] investigated singular and regular perturbation methods as applied to the natural frequencies and mode shapes of a rotating beam.
The rotating beam system is often modelled as an Euler-Bernoulli beam with clamped-free or pinned-free boundary conditions. The classical model neglects the effect of any payload, and also that of the hub inertia on the system dynamics. To demonstrate the use of variational calculus for the slewing beam system, Morris and Taylor [10] provided a detailed derivation based on a clamped model studied in references [11][12][13].
Recent works [5,[12][13][14][15][16]] addressed an importance issue, that of selecting different sets of modes for problems of elastic beams that undergo large rigid-body displacements. Experimental results [5,14] suggested that the exact natural frequencies are intermediate between the clamped and pinned cases. Shabana [16] demonstrated that different mode shapes that correspond to different sets of natural frequencies can be used to obtain the same resonance conditions by using simple co-ordinate transformations. Bellezza et al. [12] developed a mathematical model for a flexible slewing beam, in which two formulations with clamped and pinned roots are each modelled and compared. The frequency equation of the rotating loaded-beam system was also generated.
The objective of this note is to re-tackle the issue of boundary conditions in the modelling of slewing beam systems, by taking account of those results given in some recent works. A rotating beam driven by a motor and carrying a payload is considered in this note. Analytical frequencies by virtue of the developed model are compared with those obtained experimentally.
2. ๏ญ๏ฏ๏ค๏ฅ๏ฌ ๏ฃ๏ฏ๏ฎ๏ณ๏ฉ๏ค๏ฅ๏ฒ๏ฅ๏ค ๏ก๏ฎ๏ค ๏ฉ๏ด๏ณ ๏ฆ๏ฒ๏ฅ๏ฑ๏ต๏ฅ๏ฎ๏ฃ๏น ๏ฅ๏ฑ๏ต๏ก๏ด๏ฉ๏ฏ๏ฎ
The considered model is made of an actuator at the base with total (rotor and hub) inertia J 0 , a flexible beam with uniform linear mass density rA, length l and a payload