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Edited by: Seung-Bok Choi, Inha University, South Korea

Reviewed by: Xiaomin Dong, Chongqing University, China; Xufeng Dong, Dalian University of Technology (DUT), China; JinHyeong Yoo, Naval Surface Warfare Center Carderock Division, United States

This article was submitted to Smart Materials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Inerters are two-terminal mass elements in which the forces applied at the terminals are proportional to relative acceleration between the nodes. The volume and weight of inerters are much smaller than those of any conventional mass element for the same force, which is beneficial for engineering applications. The inerter in mechanical systems corresponds completely to the capacitor in electrical systems, which makes it more convenient to do related investigations based on mechanical-electrical analogies. A semi-active inerter (SAI) featuring a magnetorheological (MR) effect with tunable inertance is proposed, designed, and investigated to enhance the performance of the passive inerters. The proposed SAI consists of a flywheel, a flywheel housing, a ball screw, a connection sleeve, bearings, upper and lower covers, excitation coils, and MR fluid. MR fluid fulfilled in the flywheel housing of the SAI is energized by the excitation coils with applied current, and correspondingly the mechanical characteristics of the SAI are tunable via the applied current. The mathematical model and the mechanical performance of the SAI are established and tested, respectively. The nonlinearity of the experimental results is analyzed and the non-linear model of the SAI is further established. The preliminary principle verification of the continuous adjustment of the equivalent inertance of the SAI is conducted using the non-linear model. Moreover, a compensator is proposed to address the problem of the phase difference between the controllable force and the real output force of the SAI, and continuous inertance adjustment of the SAI with a compensator is realized.

In Smith (

At present, four inerter types can be found: rack pinion type (Smith,

The inertance of the conventional inerter is not tunable and the bandwidth of vibration suppression of the conventional inerter-based system is narrow in turn, which would be a restriction for inerter applications. Hu et al. (

Structural principle of the SAI.

where _{b} is the output force of the conventional inerter; _{1} and _{2} are the displacements at the two ends of the inerter.

Ideal model:

According to equations (1) and (2), the ideal model of the conventional ball screw-type inerter is only related to the characteristics of the flywheel and the ball screw. The output force of the proposed SAI is dependent on the controllable force due to the viscosity change of the MR fluid. It can be expressed as:

where _{MR} is the controllable force due to the viscosity change of the MR fluid.

_{MR} is expressed as:

where _{vis} is the viscous torque produced by the MR fluid in field-off state (i.e., no applied current in the excitation coils) and _{MR} is the field-dependent torque. _{vis} and _{MR} are respectively given by:

where η is the viscosity of the MR fluid in field-off state; τ_{y} is the shear yield stress of the MR fluid; _{1} is the outer radius of the flywheel; _{2} is the inner radius of the flywheel housing; _{d} is the rotational speed of the flywheel; _{d} is the width of the annular gap; _{d} is the axial length of the flywheel; δ is the correction coefficient with consideration of the influence of the magnetic leakage and ξ is the coefficient of the effective area (i.e., the ratio of the effective area to the ideal area) (Bai et al.,

The inertance is defined by the ratio of the output force of the inerter to the relative acceleration between its two ends. That is, the corresponding inertance can be obtained by the relationship between the output force and the relative acceleration of the inerter. The output force of the proposed SAI is controllable and related to the applied current. When under a certain relative acceleration, the desired inertance can be achieved by adjusting the applied current. The inertance obtained by adjusting the applied current is defined as the equivalent inertance

Practically, the movement of the inerter should overcome a certain amount of inherent friction. Therefore, the output force

where

The prototype of the SAI is fabricated to verify the objective mechanical properties. As shown in

Experimental setup for the SAI.

When the applied current is 0.5 A, the output force of the SAI under sinusoidal displacement excitations with a frequency of 0.25 Hz and different amplitudes:

When the applied current is 0.5 A, the output force of the SAI under sinusoidal displacement excitations with an amplitude of 20 mm and different frequencies:

When the applied current is different, the output force of the SAI under a sinusoidal displacement excitation with a frequency of 1.0 Hz and an amplitude of 20 mm:

Based on the comparison of

As shown in

Nonlinear model of the SAI.

In the experimental tests, one end of the SAI is fixed, as shown in _{3} = 0. According to

where _{s} and _{s} are the stiffness and damping of the ball screw, respectively.

The minimal inertance of the proposed SAI is set to 600 kg (i.e., the inherent inertance is 600 kg when the applied current is 0 A). When the SAI is excited by sinusoidal displacement excitations with low frequencies and no current is applied, the output force is dominated by the inherent friction force, and the inertia of the flywheel can be ignored. In this case, the output force of the SAI can be approximately regarded as the inherent friction force. In this study, a sinusoidal displacement excitation with a frequency of 0.1 Hz and an amplitude of 10 mm is applied to the SAI in field-off state and the inherent friction force

Parameter identification of the nonlinear model of the SAI is conducted using experimental data under different operating conditions. The parameters _{s}, _{s} and _{MR} are selected to be identified using the following criterion function with the least square method:

where _{th} is the predicted value of the output force calculated by the nonlinear model of the SAI and _{exp} is the experimental value of the output force of the SAI.

The stiffness _{s} of the ball screw is 5,500 kN/m and the damping _{s} is 7,000 Ns/m according to the parameter identification. The controllable force _{MR} changes with the applied current, which is consistent with the actual situation.

Comparison between the experimental and predicted results when under a sinusoidal displacement excitation with an amplitude of 20 mm and a frequency of 1.0 Hz and a 0.5 A applied current.

The established nonlinear model of the SAI can be used to achieve adjustment of the inertance. The SAI is separately subjected to an adjustment simulation of the 0–6,000 kg inertance under the condition of the lower and upper bound force (about 0–4 kN according to the test results) of the SAI.

The equivalent inertance _{MR} under a sinusoidal displacement excitation

The output force (inertance) of the SAI is related to the acceleration, while the controllable force is related to the velocity (according to Equations 4, 5). There is a 90° phase difference between the output force and the controllable force of the SAI, which may result in failure to achieve the desired inertance when adjusting the controllable force. However, the continuous adjustment of the equivalent inertance would be achieved if a force compensator (i.e., phase adjustment mechanism) could be provided.

The SAI with a compensator is proposed and shown in

Schematic of the SAI with a compensator.

Adjustment principle of the equivalent inertance.

The output force of the SAI with a compensator can be expressed as:

where _{com} is the force generated by the compensator.

The equivalent inertance _{MR} and the generated force of the compensator _{com} under a sinusoidal displacement excitation

The equivalent inertance of the ball screw-type SAI, as shown in

where _{1} is the input displacement of the payload _{2} is the displacement of the payload

A single-degree-of-freedom vibration control system with a SAI:

_{1} = 310 N, _{2} = 110 N and _{3} = 10 N when the SAI has no compensator. As shown in

Acceleration response of the single-degree-of-freedom vibration system under a sinusoidal displacement excitation with an amplitude of 50 mm and a frequency of 1.0 Hz.

In order to enhance the performance of the conventional passive inerter, the structural principle of an SAI was proposed and studied in this paper. It is composed of a flywheel, a flywheel housing, a ball screw, a connection sleeve, bearings, upper, and lower covers, excitation coils, and MR fluid. The proposed SAI achieves adjustment of the inertance by adjusting the applied current in the excitation coils and has the advantages of a simple structure and wide adjustable range. The mathematical model of the SAI was established and the mechanical properties of the SAI were tested based on the experimental setup. The test results indicate that the nonlinear factors of the ball screw cannot be ignored for mechanical performance description of the SAI. A nonlinear model of the SAI was established and the parameters were identified by the least squares method. The continuous adjustment of the equivalent inertance was realized by integrating a compensator to overcome the phase difference between the controllable force and the output force. Vibration attenuation performances of a single-degree-of-freedom vibration system based on the SAI with and without a compensator, as a very preliminary application case, are analyzed and compared. Based on the research of this paper, the conclusions are summarized as follows.

(i) The established nonlinear model of the SAI can effectively describe and predict the mechanical properties of the SAI.

(ii) The problem of the phase difference between the controllable force and the output force can be solved via a compensator. The continuous adjustment of an equivalent inertance with a range of 0–6000 kg could be achieved by adjusting the applied current. The proposed and employed compensator presents a helpful approach for the phase control of force in mechanical systems.

(iii) For a single-degree-of-freedom vibration system, the SAI with no compensator still provides a better vibration isolation performance than the conventional system, and the smaller the inherent friction force is, the better the performance will be.

W-MZ carried out the modeling, computation and experimental work, and helped draft the manuscript. X-XB conceived the conception, investigated the technical background, designed, and coordinated the study, and drafted and revised the manuscript. CT helped do modeling and experimental work and draft the manuscript. A-DZ helped draft and revise the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors wish to acknowledge the Key Research and Development Projects of Anhui Province (Grant No. 1704E1002211) for her support of this research.