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A vibration control method based on energy migration is proposed to decrease vibration response of the flexible arm undergoing rigid motion. A type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, vibration energy of the flexible arm can be dissipated by not only the damping of the vibration absorber but also its own enhanced damping, thereby effectively decreasing vibration. Through numerical simulations and analyses, it is proven that this method is feasible in controlling nonlinear vibration of the flexible arm undergoing rigid motion.

Although flexible robotic arms have various important applications in space exploration, automatic assembly, undersea operation, nuclear environment, and so forth, major possible disadvantages of these arms are serious vibration response and deteriorative tracking accuracy due to large dimension, light weight, low structural damping, and small stiffness. Therefore, a great deal of research has been conducted to combat these problems.

Some traditional methods have been used to control vibration, like enhancing the stiffness of links and joints, optimizing shape and dimensions [

Internal resonance is a typical nonlinear principle. Through modal interaction, vibration energy of one mode can be transferred to another mode which is commensurable or nearly commensurable with the former [

In recent years, the above studies have been extended to control vibration of the distributed flexible beam. Pai et al. [

To the best of our knowledge, there is little research on controlling vibration of the flexible arm based on energy migration. In this paper, a type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Furthermore, due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, the damping of the vibration absorber can be mapped into the flexible arm, thus increasing the damping effect of the flexible arm. In this way, vibration energy of the flexible arm can not only be dissipated by the damping of the vibration absorber based on the internal resonance but also be attenuated by its own enhanced damping, thereby effectively decreasing vibration.

In this study, a model with one flexible arm, one rigid joint, and a vibration absorber is considered, as shown in Figure

Model of the flexible arm with a vibration absorber.

Based on the assumed-modes theory, the deformation of the flexible arm can be expressed as

In present study, only the fundamental mode of the arm is considered due to its most contribution to the vibration response in common cases. Equation (

The angle of the tangent of the flexible arm at

To use the absorber for reducing vibration response that resulted from the fundamental mode of the arm, the coupling effect between the fundamental mode coordinate

Equations (

The nondimensional variables

To make the damping and nonlinearities appear in the same perturbation equations, let

The time dependence

We seek first-order approximate solutions of (

Substituting (

Order (

Order (

The solution of (

In this study, because the second-order nonlinear coupling terms exist in the dynamic model, the vibration absorber is used to control vibration of the flexible arm at the 1 : 2 internal resonance condition; that is,

In the case of the 1 : 2 internal resonance, a detuning parameter

Since the determinant of the coefficient matrix of (

Substituting (

Inserting (

Eliminating

From

In order to better understand the dynamics of the system, the undamped case (i.e.,

From (

In the presence of damping (i.e.,

The Jacobian matrix of this case is

The corresponding eigenvalues are

By numerical integrations of (

To verify the above theoretical analysis, some numerical simulations are done on the conditions: ^{3}.

Supposing the flexible arm has neither rigid motion (i.e.,

Response of the uncontrolled fixed flexible arm.

A fixed flexible arm in ADAMS.

Response of the uncontrolled fixed flexible arm in ADAMS.

In order to reduce vibration of the flexible structure via modal interaction, a vibration absorber is perpendicularly attached to the flexible arm at

Undamped modal amplitudes in fixed flexible arm.

When the damping of the vibration absorber is taken into account and

Damped modal amplitudes in fixed flexible arm.

The end-effector deformation of the flexible arm equipped with the vibration absorber is shown in Figure

Response of the controlled fixed flexible arm.

Through above simulation and analysis, it is verified that this control method based on internal resonance is effective in controlling vibration of the flexible structure.

Next, an example of the flexible arm undergoing rigid motion is considered. Suppose the desired joint motion of the arm is

If the flexible arm is not equipped with the vibration absorber, when moving according to (

Response of the uncontrolled moving flexible arm.

Response of the uncontrolled moving flexible arm in ADAMS.

In order to reduce vibration of the flexible arm via modal interaction, a vibration absorber is perpendicularly attached to the flexible arm and the same parameters as in the aforementioned case are used. At the state of internal resonance, (

Undamped modal amplitudes in moving flexible arm.

When the damping of the vibration absorber is taken into account and

Damped modal amplitudes in moving flexible arm.

The end-effector deformation of the flexible arm equipped with the vibration absorber is shown in Figure

Response of the controlled arm (

Response of the controlled arm (

In addition, the effects of the absorber damping

Effects of the absorber damping.

Furthermore, the effect of the absorber location is studied. When the absorber is attached to the flexible arm at

Response of the controlled arm (

Response of the controlled arm (

Based on above simulations and analyses, it is proven that this method is effective in controlling nonlinear vibration of the flexible arm.

In this paper, a vibration control method based on energy migration is proposed to decrease vibration response of the flexible arm undergoing rigid motion. A type of vibration absorber is suggested and gives rise to the inertial coupling between the modes of the flexible arm and the absorber. By analyzing 1 : 2 internal resonance, it is proved that the internal resonance can be successfully created and the exchange of vibration energy is existent. Furthermore, due to the inertial coupling, the damping enhancement effect is revealed. Via the inertial coupling, vibration energy of the flexible arm can be dissipated by not only the damping of the vibration absorber but also its own enhanced damping, thereby effectively decreasing vibration. Through numerical simulations and analyses, it is proven that this method is feasible in controlling nonlinear vibration of the flexible arm undergoing rigid motion.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study is supported by the Pre-Research Foundation of GEH PLA (no. 9140A34030315KG18002) and the Major State Basic Research Development Program of China (973 Program) (no. 2013CB733000).