Development of a new tuned vibration absorber based on one degree-of-freedom of translational motion

Abstract The study has proposed a newly tuned vibration absorber (TVA) based on one degree-of-freedom (1DOF) of translational motion. The absorber mass takes the shape of a tank filled with fluid. The device has been attached to the fluid circulating system, which plays the role of changing the absorber mass. It fills the tank or drains it depending on the data received from the frequency force sensor. The type of TVA used is similar to simple damped TVA. A MATLAB software has been applied, and a numerical procedure was used to test the TVA and compare the performance with the other TVAs. The method is simple to carry out the performance of the TVA as it is adequate for real-life problems.


Introduction
A conventionally tuned vibration absorber (TVA) is a single-degree-of-freedom (SDOF) system used to attenuate a structure's vibration at a particular forcing frequency. The SDOF-TVA is a vibration device that has received the attention of many researchers, which was initially proposed by Frahm about a century ago (Frahm, 1909) to configure a mass-spring system. The damper has been introduced in the TVA by Ormondroyd and Den Hartog (Ormondroyd, 1928) as they show its effect and how it can broaden the range of frequency. Different aerospace, automotive, and civil engineering sectors have ABOUT THE AUTHOR Dr Asiri's research activities are on the vibration control of mechanical systems. He got in 2010 a patent from KACST titled: Differential Agitator and a patent from US Patents titled: Smart Boat for Swimming Pool Maintenance and Water Safety. Dr Asiri currently teaches vibrations and control courses for undergraduate and graduate students. In addition, he published many papers on vibration analysis and modal analysis of Functionally Graded Materials using FEM. Eng. Almashhor got BSc of ME from University Tenaga Nasional, Malaysia and MSc candidate of ME in KAU, Saudi Arabia. He has worked in both industrial and construction sectors applying and tutoring. He is interested in two fields: Renewable energy, CFD modeling, and Design & Development. He got LEED and NDT certifications.

PUBLIC INTEREST STATEMENT
Due to the technological relevance of tuned vibration absorber both in the industrial and academic realms, it is still a subject of permanent interest. Devices for stabilizing ship roll motion, for enhancing the comfort of users on pedestrian bridges, and for attenuating vibrations are its new applications. The use of tuned vibration absorber mitigates the dynamic forces transmitted because of its high rates imposed on the canon motion. Generally, a tuned vibration absorber is designed for attenuating vibrations produced by an entirely harmonic excitation.
been studied for several decades since its instigation. A TVA is an auxiliary system whose parameters can be tuned for suppressing the host structure's vibration in its most generic form. The TVA suppresses the vibration at its point to the host structure using the interface force application as the auxiliary system is majorly considered a spring-mass-damper system. The tuned frequency of the TVA represents the undamped natural frequency with its base blocked.
Den Hartog (Den Hartog, 1985) introduced the two-step design technique, which starts with selecting resonance frequency and then determining optimal level damping. It is based on observing two invariant (fixed) points in the frequency response function of the TVA and the host structure. The two fixed points are the same when changing the damping. The optimal value of damping can be obtained by averaging the two damping coefficients obtained by setting the derivative at the two fixed points to zero (Brock, 1946;Den Hartog, 1985). This approach gives a good performance, but this approach gives an optimal design for specific range vibration. Many researchers have been conducted in the TVA that allows changing its stiffness to broaden the frequency range. Wu and Shao (Wu & Shao, 2007) introduced a virtual vibration absorber in which the virtual spring stiffness was tuned depending on the difference of the primary body and the acceleration based on the algorithm. Another active dynamic vibration absorber has been developed using the voice coil motor (Chen et al., 2005;Wang et al., 2016).
The vibration energy of the main vibrating system has been widely researched through TVA or TMD. Its concept is to segregate the vibration energy part of the main system throughout the secondary system. Generally, an additional mass-spring-damper system was used in TVA or TMD associated with the main system and considered active vibration control. The design of TVA is similar to that of the operating frequency of the main system. Recently, the progression of TVA emerges as an active and adaptive TVA, additionally to a passive TVA. Den Hartog (Den Hartog, 1985) initially introduces the fundamental concept of passive TVA while investigating the method of reducing the influence of ship rolling. It has been observed that many problems to artificial structures and devices are rose due to vibrations. It can be annoying for individuals in a vehicle or a building, or it might even lead to structural collapse or metal fatigue. The least expensive and most efficient way to reduce vibration issues is to instigate damping, a mechanism for mitigating the energy from the vibrations (Bonello, 2011). There are several advantages of using TVA, such as easy mounting, reducing low-frequency band, and simple design. TVA is effectively used at a specific-defined operating frequency as an anti-resonance (Tophøj et al., 2018). When the operating or excitation frequency is constant, the best performance of TVA is shown in reducing vibration. In contrast, the performance of undamped TVA is low when the operating frequency is differentiated. A study has developed a damped TVA to reduce vibration of the main system in a wide frequency range (Bonello, 2011). Another study has instigated an adaptive vibration absorber for minimizing transient vibrations and steady-state vibrations (Weber et al., 2016). To validate the control and robustness performances, a study uses a hybrid dynamic vibration absorber of multi-degree freedom structure using experimental work. Another study has shown the effectiveness of active vibration control for builders subjected to vertical and horizontal large seismic excitation (Lin et al., 2018). A study conducted on vibration control of slab breaker machines improves the effectiveness of the TVA by using passive dual-mass tuned vibration absorber (Kluger et al., 2015).
In this regard, the study has described a new type of tuned vibration absorber (TVA) one degree-offreedom (1DOF) of translational motion. The system has been mathematically and numerically modeled and simulated. The simulation results were experimentally verified and analyzed. Due to the variation of the vibration source distance and TVA distance from the COG, this model was appropriately utilized for simulating the vibration responses and vibration reduction of the 1DOF main system. The type of TVA used in this study was similar to the normal spring-damper mass system, but tuning will be the act of changing in mass and not changing in stiffness. The study has examined the derivative control for damped vibration absorbers and proportional control for undamped vibration absorbers. In the case of an undamped vibration absorber, the impact of active control on the resonant frequencies was presented. This paper has investigated a supported beam with 1DOF of different metrics for gaining physical insight into the mechanisms of vibration control. Lastly, the findings were summarized that develop the core attributes of such systems for vibration suppression.

Materials and methods
In this study, the MATLAB Simulink has been used to create a simple model of an SDOF with the translational move from the basic equations. The TVA has been studied initially by deriving the equations of the system as the exciting force was applied on a primary mass. Afterward, a procedure design was conducted to find out the parameters of the TVA. Finally, the effect of changing the absorber mass was carried out. However, the idea of a variable massed system has been presented later.

Model
The TVA used in this study is shown in Figure 1. The TVA consists of spring and damper in parallel ka, ca, and mass ma, connected to a host structure of mass m1 and stiffness k1 using the spring and damper. A circulation system has been attached to supply and drain the fluid, which effectively changes the absorber mass. The TVA may translate vertically as the host structure was subjected to base excitation to be moved in the vertical direction.

Analysis
The distance ratio was differentiated from 0 to 4.24 by shifting the absorber cantilever from the beam gravity to the right side. Excitation's frequency varies from 0 Hz to 30 Hz. A free body diagram of the main system was used to obtain the equations.
The equations of motion are expressed in the following paragraphs based on an active control system with proportional gain for controlling mass vibration.
The equation of motion will be: Here, the actuator was devised for exerting a controlled force F c such that F c ¼ À k p x 2 . Moreover, the steady-state amplitude of masses was obtained via harmonic solution x j ¼ X j sinωt; j ¼ 1; 2.
Equation 4 gives the following outcome when the amplitude of machine mass is zero; The amplitude of machine operating will be zero at its original resonant frequency using the undamped dynamic absorber. Moreover, the natural frequencies of core mass and absorber mass links with an operating frequency in the form of; Two resonant frequencies were introduced in the dynamic vibration absorber whereas it eliminates the vibration at operating frequency ω, as well as the operating frequency ω, must be kept away from the two resonant frequencies Ω 1 Ω 2 . The denominator of equation 5 was adjusted to zero after setting the values of two resonant frequencies Ω 1 Ω 2 . The roots obtained were denoted in the square of non-dimensional resonant frequencies with absorber mass by setting the denominator of equation 5. The resonance equation has been analyzed to Where: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The vibration was eliminated by the active undamped dynamic vibration absorber mentioned in the prior section at operating frequency ω but instigated two resonant frequencies Ω 1 Ω 2 . An active damped dynamic vibration absorber reduces the amplitude of the machine.

Parameters
The selection of absorber damping depends on absorber spring and mass, specifically on its machine or structure that needs to attenuate the vibration. The following parameters were selected for the simulation results.

Results and discussion
In the analysis of active undamped vibration absorbers, the parameters undertaken are the gain for absorber mass stiffness. Findings of differences in resonant frequencies of one degree of freedom are compared for different parameters of the proportional gain ratio of freedom main and absorber mass. The codes and the Simulink model were shown in the Appendix. For the first experiment, the natural frequency was constant while the exciting force was applied on the primary mass, where the changes in frequency force effect were shown in Figure 2. By putting a value for ca, the vibration changes, as shown in Figure 3. It has been observed that the variation between two forces elevates with an increase in mass ratio µ. It was observed that r 1 is fewer than and r 2 is greater than the machine's operating speed.
The working of TVA was effectively observed in unique structures that possess similar natural frequencies with a specific force-frequency range. To use this absorber in different structures, a change in absorber spring or mass was applied. Since this paper focused on the mass, the absorber spring will be set as constant, although a circulating fluid system was attached. Figure 4 shows the effect of changing mass while the frequency force was equal to the natural frequency.
There was a low peak when changing the absorber mass where it should fall in. When the fluid fills in the tank, the mass will change as long as the weight. In response to the other conditions of frequency force, the ratio of w/wa has been studied in Figure 5. Figure 5 shows that the increase or decrease in the absorber frequency is equal to the forcefrequency, attenuating the vibration located on point 1 over the w/wa axis. Another case was studied when all the three frequencies (Natural, Force, and Absorber) were equalized, and a sudden change in force-frequency occur. Figure anFigure 6d Figure 7 show clear low peaks where the absorber frequency must fall in. These figures were obtained from a sensor attached to the system.
The result of the Simulink model was compatible with the previous results where parameters from Table 1 were inserted, and the force-frequency was set to be equal to the natural frequency. Figure 8 presents the resonance concerning time in which the absorber frequency was computed to be less than the force-frequency.
The sensor must measure the resonance, and the circulation system must reduce the mass by draining the fluid using the drain valve. Reducing the absorber mass will increase its frequency, and it will stop draining when the two frequency matches. Figure 9 shows the final result. From Figure 8 anFigure 8d Figure 9, it was obvious that the effect of the TVA reduced the peak of the resonance from 0.8 × 10-3 to 0.2 × 10-3.  Mass ratio, damping ratio, the ratio of derivative gain to damping consent of absorber mass, and the ratio of natural frequencies were the metrics undertaken to investigate active damped dynamic vibration absorber. Findings of non-dimensional steady-state reaction were compared with different metrics of mass ratio and derivative gain ratio. It can be seen that one degree of freedom was the most efficient vibration for which the ordinates of two peaks' points were equal. Based on these observations, an absorber fulfilling this feature was an optimally tubed active vibration absorber. The core values were established by developing the resonant response as flat as possible at the two peaks.

Conclusions
In this study, an SDOF TVA has been developed by adding the variable mass system, using the circulation fluid system to change the mass as per the system required. A model graph was extracted using MATLAB software. It was shown from the graphs that by increasing or decreasing the force-frequency, a low peak was developed in the resonance graph. Therefore, to attenuate the vibration, the frequency ratio must fall in that low peak. In particular, reducing the force-frequency will reduce the absorber frequency and require adding fluid to absorber mass. The size of the tank (absorber mass) can be obtained by the less force-frequency when applied to the system. The tank weight (empty) should be applied based on the maximum force.
Further research is needed to study the time required for the system. The presented control algorithm was devised for prioritizing the frequency control of the TVA 1DOF by precise stiffness emulation as damping and stiffness emulations were computed with semi-active damper quantities. Thereby, it was essential for future studies to emphasize the progression of damping and stiffness correction strategies for improving the damping emulation accuracy to maintain precise stiffness emulation.

Data availability
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Disclosure statement
The author declares that there is no conflict of interest regarding the publication of this paper.