Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam

https://doi.org/10.1016/j.jsv.2007.10.044Get rights and content

Abstract

In this paper, two methods for controlling the residual vibrations of a translating or rotating Euler–Bernoulli cantilever beam are considered. Although a beam has an infinite number of vibration modes, when it simply changes its position by translation or rotation the first mode is the main contributor to the total response. Thus, the problem can be reduced to the base acceleration excitation of a single-degree-of-freedom system. Two simple methods are suggested for suppressing the residual vibration of such a system without considering any control algorithms. Both methods are based on the transient response of the system—namely, the shock response spectrum (SRS). The first method is simple and can be used for lightly damped systems, while the second method can be applied to more general situations. The result of the second method is similar to that of the input shaping method; however, in the method proposed here, both position and time to move from one position to another can be controlled simultaneously.

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 byy¨(t)+ωn2y(t)=-u¨(t).

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.

References (13)

There are more references available in the full text version of this article.

Cited by (26)

  • An improved vibration control method of a flexible non-uniform shaped manipulator

    2021, Simulation Modelling Practice and Theory
    Citation Excerpt :

    Free vibration of a rotating beam with a nonlinear spring-mass system is investigated and the natural frequencies of the system are given in the reference [31]. Two methods based on the shock response spectrum are presented in the reference [32] to eliminate the residual vibration amplitudes of a rotating flexible beam. The researchers showed zero amplitudes of residual response by using appropriate pulse duration.

  • Effect of drive control on dynamic characteristics of spacecraft tracking-drive flexible systems

    2021, Mechanical Systems and Signal Processing
    Citation Excerpt :

    The dynamic problem of the STFS was simplified into a problem of the rigid-flexible motion of the flexible appendages, which were rotated by an external drive torque, and thus the dynamic characteristics of the STFS corresponded to those of the flexible appendages. Models of the STFS can be divided into two categories based on the modeling method of the flexible appendages; the first type includes discrete system models, in which the flexible appendages are modeled using the lumped parameter method [14–17], modal coordinate method [18–20], or finite element method [21,22]; the second type includes continuous system models, in which the flexible appendages are represented as cantilever beams [23–25] or clamped-free-free-free Kirchhoff plates [26]. However, the STFS includes not only the rigid-flexible coupling inherent to the rotation of the flexible appendages but also the dynamic interaction between the drive mechanism and flexible appendages, as well as the servo drive control inside the drive mechanism.

  • Dynamic analysis on hub–beam system with transient stiffness variation

    2019, International Journal of Mechanical Sciences
  • Vibration suppression of rotating long flexible mechanical arms based on harmonic input signals

    2018, Journal of Sound and Vibration
    Citation Excerpt :

    The open-loop control methods do not have such high requirements on the model and do not need additional sensors. Among them, the input shaping is the most widely studied one ([22–25]). Although many variants exist for the input shaping method, the basic principle is consistent: the input shaping signal is the convolution of the initial input signal and a series of pulse signals.

  • Intelligent tuning of vibration mitigation process for single link manipulator using fuzzy logic

    2017, Engineering Science and Technology, an International Journal
View all citing articles on Scopus
View full text