Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam
Introduction
When a flexible structure changes its position by suddenly translating or rotating following a command input, residual vibration is inevitable due to the inertial load imposed on the flexible structure. Suppressing this type of residual vibration has become very important in many engineering applications such as space structures, cranes and flexible robot manipulators. There are two main categories for controlling the residual vibrations; one is closed-loop control, for example PD, PID, and adaptive control [1], [2], [3], [4], and the other is open-loop control, using methods such as pre-shaping the command input [5], [6], [7], [8], [9]. The latter method has been applied widely since being suggested by Singer and Seering [5]. It can be implemented easily once the dynamics of the structure, namely the natural frequencies and damping ratios, are known. Although there are many variants of the method to enhance the robustness, the basic principle is the same, which is to design the best filter to suppress the residual vibrations. For the input shaping method, the filter consists of a series of impulses (Fig. 1).
The aim of this paper is to show that the same results can be achieved with only knowledge of the system dynamics, without considering the filter, i.e., we do not rely on any control strategy. Two methods are proposed, one is simple and fast and the other is somewhat related to the input shaping method. Both methods are purely based on the transient response of the structure—e.g., the shock response spectrum (SRS), and are much easier to implement because there is no need to consider a control algorithm. Moreover, the proposed methods control both the position and the time taken to change the position simultaneously. The basic idea follows from the SRS [11], [12]. For an undamped single-degree-of-freedom system with base excitation, when the excitation force is a pulse-like input (e.g., rectangular, half-sine, etc.), the residual response is zero if the duration of the pulse is appropriately chosen. If this principle is applied to our case, then the problem becomes to determine an input (pulse) that automatically suppresses the residual vibration by considering the response of the structure excited by a pulse-like input. The problem can be depicted as in Fig. 2.
A typical application of the method may be a slewing flexible robot arm that follows a command input signal. Although there are many modes of vibration, for this particular problem, it is generally sufficient to consider the first mode only. This is demonstrated in Section 2. The details of the new approach follow in subsequent sections.
Section snippets
Response of a flexible beam under an inertial force
Consider a flexible robot arm crudely modelled as a uniform cantilever beam rigidly attached to a rotating hub as in Fig. 3, where the hub rotates to a desired angle following the command input. In this model, inertia of the hub is neglected. As in the references, for example [1], [3], [6], [9], [10], we neglect the centrifugal force, which may arise due to the rotation of beam. (Note that we do not consider the vibration while the beam is rotating, but the main concern is the ‘residual’
Control of residual vibration using the transient response method
As mentioned earlier, the methods described in this section are closely related to the SRS [11], [12]. From the SRS of a single-degree-of-freedom system it is possible to determine the shock duration such that the residual amplitude of the SRS is zero for a given shock pulse shape. Because the first mode of a rotating beam is of interest, a single-degree-of-freedom system subject to base acceleration as shown in Fig 4 is considered. The equation of motion is given by
Now the
Conclusions
Two simple methods have been proposed to suppress the residual vibrations of a translating or rotating flexible cantilever beam excited as it is moved from one position to another. The methods are based on the transient response of the system, i.e., the SRS, and do not require any filtering processes or control algorithm. Once an appropriate velocity profile has been chosen by considering the SRS or the Fourier transform of the velocity pulse, the methods can be applied in a straightforward
Acknowledgements
The research was supported by a grant from the 2006 International Academic Exchange Program of Andong National University.
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