This study proposed a novel hybrid dynamic balancer and vibration absorber that is cheaper than active dampers and more effective than passive dampers. The proposed damping system does not need to be altered structurally to deal with different damping targets. Rather, the proposed vibration absorber is capable of selfadjustment to the optimal damping location in order to achieve balance and, thereby, optimize damping effects. The proposed device includes a groove under the damping target with inertial mass hung from a coil spring beneath. This allows the device to bounce vertically or rotate in order to reduce vibrations in the main body. The coil spring vibration absorber can also slide along the groove in order to adjust its location continuously until the vibrations in the system are minimized and the main body is balanced. Experiments verify the efficacy of the proposed device in improving damping performance beyond what has been achieved using conventional devices. We also provide an explanation of the theoretical underpinnings of the design as well as the implications of these findings with regard to future developments.
Vibration can affect the stability of a structure, and constant vibration can lead to fatigue and structural damage. Babitsky and Veprik [
Vibration damping methods can roughly be divided into two types: active and passive. Generally speaking, active damping is the most effective approach to vibration damping; however, most conventional machine tools vibrate in the vertical direction, which is best dealt with using a tunedmass damper (TMD) comprising a mass and one or more springs. The first theoretical investigation of TMD was performed by Den Hartog in 1947 [
In general, TMDs or dynamic vibration absorbers (DVAs) are resonant devices used to damp or absorb vibration. In particular, the DVAs are undamped absorbers. Heo et al. [
Wang and Chen [
All of these studies involve one or more TMDs (or DVAs) placed at fixed locations. They present optimal damping effects under fixed external forces; however, they are not necessarily effective in dealing with external forces that change or for rotating inplane vibrations. ABS applications provide a track to guide the movement of the damper at balancing position; however, they are unable to provide exert optimal damping effects on the vertical vibrations in general machine tools or outofplane vibrations. This study designed a hybrid dynamic balancer and vibration absorber (HDBVA) to reduce the vibrations in operating equipment with rigid structures. The proposed vibration absorber is simple in structure, costs less than active vibration absorbers, and need not be structurally altered to deal with variations in external forces. The vibration absorber moves along a groove when changes occur in the operating environment. The vibration absorber also balances instabilities in the main body and automatically moves to the location required for optimal damping, thereby achieving vibration reduction superior to that afforded by passive vibration absorbers. This innovation eliminates the shortcomings of existing active and passive dampers, combines the advantages of both, and elevates vibration reduction to a new level.
The main body in this study comprises a rigid plate, the four corners of which are fixed to an optical table using springs. Computer numerical control (CNC) was used to cut a groove on the panel to guide the movement of the vibration absorber. The application of external forces to the main body was achieved using a signal generator to send a sine wave to an actuator, which then applied the force to the rigid plate. A laser displacement gauge was used to measure vertical vibrations (
This section describes the development of the equations of motion using Lagrange’s equation. We first defined all of the symbols and their relative positions in the system, as shown in Figure
Theoretical model of the rigid plate with vibration absorber.
As for the vibration absorber,
Using the above derivation process, we can obtain a 3D equation of motion for the rigid plate and vibration absorber system. As for the frequency responses in the system, we assume that
The above derivation process produced the equation of motion for the rigid plate and vibration absorber system. The equation of motion for the rigid plate without the HDBVA can be obtained by eliminating the terms of the HDBVA in the theoretical model. It should be noted that, for a rectangular plate,
The analytical solutions for the undamped natural frequencies in each DOF are
Parameters in numerical simulations of rigid plate with vibration absorber.
Parameter  Symbol  Value 

Mass of lateral groove plate (kg) 

2.5 
Mass of diagonal groove plate (kg) 

2.0 
Mass of longitudinal groove plate (kg) 

1.8 
Mass of DHDBVA (kg) 

0.05 
Spring constant of the plate (kg/sec^{2}) 

13500 
Damping coefficient of plate (kg/sec) 

11.8 
Torsional spring constant of plate (kg·mm^{2}/sec^{2}) 

33000000 
Torsional damping coefficient of plate (kg·mm^{2}/sec) 

11.8 
Plate massmomentofinertia in 

4687.5 
Plate massmomentofinertia in 

1171.875 
Plate massmomentofinertia in 

5859.375 
Coil spring constant of HDBVA (kg/sec^{2}) 

2400 
Coil spring torsional constant of HDBVA (kg/sec^{2}) 

2400 
The purpose of the following experiment was to verify the accuracy and feasibility of the HDBVA. Our primary focus involved measuring the amount of vertical vibration as well as the amount of rotation around the respective axes in the rigid plate vibration model. In these experiments, we considered three metal plates with a length of 300 mm, a width of 150 mm, and a height of 20 mm. Grooves were etched in three directions for the simulations; the model with a diagonal groove is presented in Figure
Picture of the diagonal groove rigid plate model.
Sliders and inertial mass for HDBVAs.
Slider for HDBVA1
Slider for HDBVA2
Slider for HDBVA3
Inertial mass of HDBVAs
Experimental set up of HDBVA1 on lateral groove rigid plate.
Lateral groove rigid plate
HDBVA1
Schematic diagram of experimental setup.
This study measured the amplitudes and amount of rotation in each DOF corresponding to different frequencies. We then plotted frequency response graphs for each DOF and compared the simulation results with and without the vibration absorber. In the experiments, the initial locations of the sliders included the nearforce point (beneath the point at which the force was applied) and the farfromforce point (at the corner of the symmetrical quadrant) to confirm whether the vibration absorber provided optimal damping effects regardless of its initial position.
Figures
Numerical results of frequency response in
Numerical results of frequency response in
Figures
Numerical results of frequency response in
Numerical results of frequency response in
Figure
Types of groove in the main body and location of force application in experiments.
Lateral groove
Diagonal
Longitudinal
Diagonal
The previous section only presents the numerical simulation results when the HDBVA is fixed at the ends of the grooves. To achieve selfadjusting balance and vibration reduction, we measured the effectiveness of various combinations of sliders and grooves. During the experiments, we located the point of force at a corner of the plate to simulate damping effects under extreme conditions. To identify the locations in which the vibration absorber provides the best damping effects, we recorded the slid displacement of the sliders and measured the amplitudes in the various DOFs after the sliders stopped moving. We then compared the results with those of numerical simulations.
Slider displacement of HDBVA1 in lateral groove.
Hz  Starting at nearforce pt.  Starting at farfromforce pt.  

Stop Pos.  Disp./cm  Disp./%  Stop Pos.  Disp./cm  Disp./%  
5  N  0  0  F  0  0 
6  N  0  0  F  0  0 
7  N  0  0  F  0  0 
8  N  0  0  F  0  0 
9  N  0  0  F  0  0 
10  N  0  0  F  0  0 
11  N  0  0  F  0  0 
12  N  0  0  F  0  0 
13  N  0  0  F  0  0 
14  N  0  0  F  0  0 
15  N  0  0  F  0  0 
16  N  0  0  F  0  0 
17  N  0  0  F  0  0 
18  N  0  0  F  0  0 
19  N  0  0  F  0  0 
20  N  0  0  F  0  0 
21  N  0  0  F  0  0 
22  N  0  0  F  0  0 
23  N  0  0  F  0  0 
24  N  0  0  F  0  0 
25  N  0  0  F  0  0 
26  N  0  0  F  0  0 
27  N  0  0  F  0  0 
28  N  0  0  F  0  0 
29  N  0  0  F  0  0 
30  m  6.5  50.0  F  0  0 
31  N  0  0  F  0  0 
32  m  6.0  46.154  F  0  0 
33  m  5.5  42.308  F  1.0  7.692 
34  m  7.0  53.846  F  0  0 
35  m  5.5  42.308  F  2.0  15.385 
36  m  5.0  38.462  F  3.0  23.077 
37  N  0  0  F  0  0 
38  N  0  0  F  1.5  11.538 
39  N  0  0  F  1.0  7.692 
40  N  0  0  F  0  0 
41  N  0  0  F  0  0 
42  N  0  0  F  0  0 
43  N  0  0  F  0  0 
44  N  0  0  F  0  0 
45  N  0  0  F  0  0 
46  N  0  0  F  0  0 
47  N  0  0  F  0  0 
48  N  0  0  F  0  0 
49  N  0  0  F  0  0 
50  N  0  0  F  0  0 
N = nearforce, F = farfromforce, m = middleofgroove.
To further elucidate the sliding conditions of the three types of slider in the various types of groove, we divided the displacement records into two parts including inrange frequency (30–40 Hz) and outrange frequency (5–29 Hz and 41–50 Hz). The relationships between displacement percentage and the location of the slider after selfadjustment are presented in Figures
Inrange slider displacement percentages of case Figure
Inrange slider displacement percentages of case Figure
Inrange slider displacement percentages of case Figure
OutRange slider displacement percentages of case Figure
OutRange slider displacement percentages of case Figure
OutRange slider displacement percentages of case Figure
In general, the vibration absorber slides a greater distance when it starts at the nearforce point
When force was applied at the corner of the plate, the diagonal groove was perfectly situated on the node line of the main body, which prevented the vibration absorber from sliding in response to the external force. Thus, we performed an additional set of experiments with the diagonal groove, the various sliders, and the force application point situated at the end of the groove (Figure
InRange slider displacement percentages of case Figure
OutRange slider displacement percentages of case Figure
Summarizing the experiment results leads to the following conclusions based on the apparent influence of slider friction on the damping effects of the vibration absorber. First, less friction enables the slider to find the optimal damping position and even enhances damping effects. When the vibration absorber starts at the nearforce point, it frequently moves toward farfromforce point to seek a better damping position, regardless of slider type. Finally, the reduced sliding capability in cases with HDBVA1 prevents the vibration absorber from moving very much when it starts out at the farfromforce point. In contrast, the better balancing and optimalpositionseeking capabilities of HDBVA3 enable it to slide and adjust its location more easily. The movement of HDBVA2 is not as smooth as that of HDBVA3; therefore, its sliding performance falls between that of the other two.
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Figures
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Figures
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Frequency response of
Finally, to identify the groove and slider combination capable of providing the best damping effects, we normalized the results for comparison by dividing the natural frequency amplitudes in the experiment results by the amplitudes corresponding to the natural frequencies without the vibration absorber. The results displayed in Table
Damping effects of groove and slider combinations (normalized).
Case of Figure 
Case of Figure 
Case of Figure 
Case of Figure 


No HDBVA Num.  1  1  1  1 
With HDBVA Num.  0.964269  1.003597  0.969740  0.908190 
No HDBVA Expt.  0.940905  0.990377  0.951089  0.884846 
HDBVA1F  0.882411  0.954460  0.908507  0.875365 
HDBVA1N  0.923682  0.960480  0.951822  0.914288 
HDBVA2F  0.852047  0.936689  0.869267  0.840238 
HDBVA2N  0.896569  0.934389  0.921326  0.871497 
HDBVA3F  0.821092  0.889291  0.840896  0.795235 
HDBVA3N  0.860223  0.898586  0.889299  0.863010 
Damping effects of groove and slider combinations (away from natural frequency and normalized).
Case of Figure 
Case of Figure 
Case of Figure 
Case of Figure 


No HDBVA Num.  1  1  1  1 
With HDBVA Num.  0.891357  1.078969  0.953960  0.778815 
No HDBVA Expt.  0.893914  0.964573  0.831144  0.756950 
HDBVA1F  0.686620  0.920159  0.834284  0.695305 
HDBVA1N  0.748457  1.048351  0.834214  0.736764 
HDBVA2F  0.759010  0.974331  0.736906  0.733996 
HDBVA2N  0.683924  0.709284  0.806611  0.682448 
HDBVA3F  0.681466  0.948458  0.812164  0.662156 
HDBVA3N  0.711264  0.862596  0.821613  0.708150 
In summary of the various graphs, the fact that the vibration absorber slides to different locations when subjected to forces at different points proves that the proposed design is more flexible than a fixed vibration absorber by enabling selfadjustment to achieve optimal damping according to the working environment. Moreover, a diagonal design with the groove passing through the force application point was shown to be the best damping design. The friction produced by the sliding of the vibration absorber also has significant influence on the capacities of the vibration absorber with regard to damping and seeking the optimal damping position. Reducing the friction provides the slider with greater freedom to move within the groove, which enhances the sensitivity of the vibration absorber in locating the ideal damping location. Furthermore, it can prevent the occurrence of more severe vibrations when the vibration absorber reaches its natural frequency.
The objective of this study was to design a selfadjusting, structurally simple, and inexpensive passive damping system that does not need to be structurally altered to deal with changes in the working environment. The vibration absorber in the proposed system is able to adjust its location according to changes in the applied external force in order to achieve the best damping effects. We first used analytical solutions and numerical methods to obtain the natural frequencies, amplitudes, and torsion angles of the various vibration models in each DOF. After comparing the results, we physically constructed vibration models for experimentation and compared the experiment results with those derived in the numerical simulations.
Many DOFs were affected by vibrations within the main body, and the external force was not fixed. As a result, the placement of a single vibration absorber in one corner of the rigid plate would not be an optimal design as it would only be able to reduce the amplitudes in a single DOF. The hybrid vibration absorber proposed in this study is able to adjust its location in response to changes in the working environment, thereby achieving superior damping effects in all DOFs by combining the advantages of active and passive vibration absorber. We also discovered that sliders with smaller friction provide the best damping results by facilitating positionseeking capabilities. In conditions with only a single vibration absorber and changing external force, designs that enable the vibration absorber to move freely along a groove provide better damping effects than those in which the vibration absorber is fixed. A diagonal design with the groove passing through the force application point and the vibration absorber starting at the farfromforce point constitute the best damping design.
The authors declare that there is no conflict of interests regarding the publication of this paper.