Design,modelling and evaluation of a variable inertance bypass fluid inerter

Abstract

The inerter emerged as a mechanical analogy to the electrical capacitor, completing the force-current analogy. It operates as a one-port, two terminal device, where the equal and opposite forces at its terminals correlate with the relative acceleration between them. This relationship is governed by ‘inertance’, a quantity that bears the unit of mass, allowing inerters to exert inertial forces. Inerters have gained considerable traction, particularly in vibration control applications. Derived from their passive counterparts, variable inertance inerters enable active control of their inertance through integrated control mechanisms. This work presents the design, modelling and evaluation of a variable inertance inerter prototype dubbed the ‘Variable Inertance Bypass Fluid Inerter’ (VIBFI). An experimental prototype of the concept was designed, constructed and tested. Simultaneously, an effort to develop and validate a mathematical model of the VIBFI is thoroughly documented. Experimental results demonstrate the controllability of performance parameters of the device, including inertance and damping coefficients, through modulating the flow restriction of the bypass channel. The mathematical models derived for the device can serve as an estimate for its performance parameters, though further refinement is required.

1. Introduction

Proposed in the early 2000s, the Inerter is a one-port, two terminal mechanical device that exerts inertial force proportional to the relative acceleration of their terminals. This proportional relationship is governed by ‘inertance’, which is a constant that carries the unit of mass (Smith, 2002). An inerter stores kinetic energy when a load causes relative motion between its terminals. Upon the release of this load, the stored energy gets released, inducing motion at the inerter’s terminals, enabling them to exert inertial force. The inerter is considered as a mechanical equivalent of the electrical capacitor to complete force-current analogy (Smith, 2002). Inerters exhibit negative stiffness and have mass amplification capability. The negative stiffness exhibited by an inerter is proportional to its inertance (Ma et al., 2021). These properties enable inerters to lower the natural frequencies of mechanical systems (Ma et al., 2021) or exert the inertial force of a large mass using a lightweight device (Smith, 2002).

Inerters can be physically embodied by coupling a transmission mechanism into an inertial energy storage mechanism. An example of such embodiment can be illustrated through a rack and pinion inerter, where a rack and pinion transmission mechanism is coupled to a flywheel for kinetic energy storage (Smith, 2002). Over the last two decades, various other embodiments of the inerter have been emerged in literature. They include, but are not limited to, ball-screw inerter (Papageorgiou et al., 2009), gear pump (hydraulic) inerter (Wang et al., 2011), helical fluid inerter (Swift et al., 2013), living hinge inerter (John and Wagg, 2019) and electromagnetic inerter (Gonzalez-Buelga et al., 2015). Each configurations have their unique strengths and limitations, making them suitable for specific applications. Realistic devices can only approximate the behaviour of the ideal inerter if they satisfy the following conditions: have small physical mass relative to its the inertance; be detachable from physical grounds; have specified linear travel and adherence to the force-acceleration proportionality relationship even when the direction of terminal motion is reversed (Smith, 2002, 2020). Realistic inerters suffer from nonlinearities such as dry friction, viscous damping, backlash and material compressibility or elasticity (Chillemi et al., 2023; Li et al., 2012; Ma et al., 2021; Wang and Su, 2008). These nonlinearities can deviate actual performance from ideal models and negatively affect performance in various applications (Brzeski and Perlikowski, 2017; Gonzalez-Buelga et al., 2017; Wang and Su, 2008).

The inerter was originally developed to be a component of the suspension network of high-performance vehicles. Incorporation of the inerter in vehicle suspension systems improves handling, ride comfort and suspension travel leading to an enhanced performance in the vehicle (Smith and Wang, 2004). Since its introduction, applications of the inerter have continuously expanded. Vibration control is a popular application of the inerter. Inerter-based vibration control systems modulate the inertial properties of the structure in addition to the stiffness and damping properties, resulting in better performance compared to conventional vibration control systems with only stiffness and damping components (Ma et al., 2021). Inerter-based vibration control systems have been studied in various applications, which includes but are not limited to structural vibration control (Ma et al., 2021), railway vehicle suspension (Wang et al., 2009), steering systems (Evangelou et al., 2007) and landing gear shimmy vibration control (Li et al., 2017).

Passive inerters feature fixed inertance, while variable inertance inerters allow adaptive control of their inertance via changing their inertance through auxiliary control systems. Chen et al. (2014) first proposed the concept of a variable inertance inerter with controllable inertance after considering the significant benefits of incorporating controllable dampers into vehicle suspension. Brzeski et al. (2015) introduced the first physical embodiment of the variable inertance inerter, integrating a continuously variable transmission between the rack-and-pinion and the flywheel of a rack-and-pinion inerter. Since its introduction, the variable inertance inerter has been employed to improve performance in structural vibration control (Brzeski et al., 2015; Hu et al., 2016; Sadeghian et al., 2021, 2022), vehicle suspensions (Chen et al., 2014; Hu et al., 2017; Li et al., 2021; Zhang et al., 2018), seat suspensions (Ning et al., 2019a, 2019b) and energy harvesting applications (Nemoto et al., 2022; Takino and Asai, 2023).

Various other configurations of the variable inertance inerter have also emerged. Based on passive ball-screw inerter, Hu et al. (2016) implemented a variable inertia flywheel to develop a variable inertance inerter. Zhong et al. (2020) implemented a magnetorheological (MR) clutch between two concentric flywheels to create a variable inertia flywheel for controlling the inertance of a ball-screw inerter. Yu et al. (2021) proposed a variable inertance gear pump inerter of which a parallel bypass channel with MR valve was integrated to control inertance. Zhang et al. (2018) proposed a variable inertance helical fluid inerter configuration which used a hydraulic cylinder-like ‘displacement-dependent inertance valve’ to control the inertance by modulating the effective length of the helical channel that the working fluid must flow through to store energy. The authors later conducted experimental verification of their device (Zhang et al., 2022). Tipuric et al. (2018, 2019) conceptually proposed a variable inertance helical fluid inerter by paralleling an MR valve with the helical channel. Its inertance can be controlled by regulating the current supplied to the MR valve. However, this design have not been prototyped and experimentally verified. Beside, the use of MR fluids also bring the concern of sedimentation for long term usage.

In light of the above motivation, this paper presents the design, modelling and experimental evaluation of a novel variable inertance inerter termed the ‘variable inertance Bypass Fluid Inerter (VIBFI)’. The VIBFI operates by directing flow through a helical channel using a hydraulic cylinder to generate inertance effects. Its inertance can be controlled by modulating the ratio of flow rates between the helical channel and the hydraulic cylinder, facilitated by a parallel bypass channel equipped with a proportional flow control valve. In comparison to existing variable inertance inerters featuring displacement-dependent inertance valves (Zhang et al., 2018, 2022), the VIBFI offers a simpler and more space-efficient design. It eliminates the necessity for a secondary hydraulic cylinder-like valve structure and the associated grounding space required for inertance control mechanisms. While sharing similarities with the proposed MR bypass inerter concept (Tipuric et al., 2018, 2019), the VIBFI utilises conventional Newtonian working fluids like hydraulic oil or water instead of MR fluids, as well as a proportional flow control valve instead of an MR valve for controlling inertance. These design choices evade common drawbacks associated with MR fluids, such as sedimentation and oxidation of magnetisable particles over time (Kumar et al., 2019). Furthermore, the experimental evaluation of this variable inertance inerter configuration addresses a gap in the literature, as previous studies have not experimentally assessed the concept of inertance control of a helical fluid inerter using a variable bypass channel (Tipuric et al., 2018, 2019).

The ultimate objective of this study is to present a configuration of the variable inertance inerter featuring controllable inertance, accompanied with a functional prototype and a reliable mathematical model that can be implemented for design optimisation and integrated into simulations of mechanical systems. To achieve this aim, the design of the VIBFI was proposed, and its mathematical model of the device was derived. Subsequently, the prototype was fabricated, and its performance was evaluated experimentally. The experimental results were then used to validate the mathematical model of the device by comparison against results simulated from the mathematical model. In this paper, Section 2 covers the conceptual design, working principles, mathematical model and prototype design of the VIBFI. Section 3 outlines the methodologies for experimental and simulation studies, Section 4 presents the results, and Section 5 discusses the results, while Section 6 concludes the paper, highlighting future work to be completed.

2. Concept and design of the variable inertance bypass fluid inerter

2.1. Concept and working principles

The variable inertance bypass fluid inerter (VIBFI) was developed from the passive helical fluid inerter, which generates its inertance effect from the circular motion of fluid in a helical channel (Swift et al., 2013). As shown in Figure 1, the VIBFI consist of a hydraulic cylinder, helical channel and a bypass channel with integrated flow control valve. Terminal 1 of the VIBFI is located on the housing of the hydraulic cylinder, while terminal 2 of the device is located on the rod end of the hydraulic cylinder. The chambers of the hydraulic cylinder are linked through a parallel network formed by the helical and bypass channels. The cylinder, the helical and the bypass channels are filled with the working fluid.

Figure 1.

Schematic of the semi-active fluid bypass inerter.

The working principle of the VIBFI is described as follows. Relative movement between the inerter’s terminals induced by applied load causes fluid to flow from one chamber of the cylinder to the other via the parallel network formed by the helical and bypass channels. The circular motion of the working fluid through the helical channel stores kinetic energy, functioning as a fluid flywheel to exert inertial forces at the inerter’s terminal. Figure 2 shows the flow direction of the working fluid when applied load to the terminals induces relative motion between the rod and the cylinder.

Figure 2.

Schematic of the VIBFI, showing the flow of working fluid.

The inertance of the VIBFI can be adjusted by controlling the flow rate ratio between the helical channel and the hydraulic cylinder. A higher flow rate ratio implies that more fluid flows through the helical channel for every stroke of the cylinder, resulting in a higher inertance, and vice versa. This control is executed by modulating the flow restriction through the bypass channel, which is achieved by controlling the opening of the flow control valve. Assuming steady flow through the hydraulic system of the VIBFI, increased flow restriction in the bypass channel leads to a higher flow rate through the helical channel. This is because the flow of fluid naturally seeks the path of least resistance in a parallel pipe network. Conversely, when the flow restriction at the bypass channel is low, more fluid flows through it, resulting in a lower flow rate through the helical channel. This variation influences the flow rate ratio between the helical channel and the hydraulic cylinder, consequently affecting the inertance of the VIBFI.

Variation of the flow rate ratio in the VIBFI not only affects the inertance, but also the damping coefficient of the device. This is because the flow of fluid from one chamber of the cylinder to another via the parallel network of channels is subjected to a pressure drop that is also dependent on the flow rate ratio. This makes the VIBFI to feature both variable inertance and variable damping within a single device. Due to the configuration of the helical and the bypass channel, there are always flow of fluid through the helical channel irrespective of the position of the flow control valve in the bypass channel. This is a fail-safe feature of the VIBFI as it enables the device to function as a passive fluid inerter even if the flow control valve is faulty or inoperative.

2.2. Mathematical model

The VIBFI is modelled as a parallel network consisting of a variable inerter a variable damper, and a dry friction component as shown in Figure 3.

Figure 3.

Mechanical network if the variable inertance bypass fluid inerter.

The output force $F(t)$ at the terminals is the sum of the inertia, damping and dry friction force, which can be presented as:

$$F\left(t\right)=b\stackrel{··}{x}(t)+c\stackrel{·}{x}(t)+f_s\mathrm{sgn}(\stackrel{·}{x}(t))$$

(1)

where $b$ is the variable inertance and $c$ is the variable damping coefficient of the device, and $f_s$ is the magnitude of static friction force. The inertance and damping coefficients are calculated from the working fluid properties, active dimensions of the VIBFI and motion of its terminals, under the assumption that the flow of fluid through the device is steady and incompressible. The inertance of the VIBFI is affected by the ratio between the flow rate through the helical channel and flow rate through the bypass channel. Let subscript $H$ denote the variable for the helical channel, subscript $B$ denotes the variable for the bypass channel and subscript $1$ denotes the cylinder parameter. Adopted from the mathematical model of passive fluid inerters (Swift et al., 2013), the inertance can be expressed in terms of $Q_H/Q_1$ by:

$$b=\rho l_H{A}_H{\left(\frac{Q_H}{Q_1}\right)}^2{\left(\frac{A_1}{A_H}\right)}^2+\rho l_B{A}_B{(1-\frac{Q_H}{Q_1})}^2{\left(\frac{A_1}{A_B}\right)}^2$$

(2)

where $\rho$ is the working fluid density, $A_i$ is the cross-sectional area of the component of subscript $i$ , $l_i$ is the length of the channel with subscript $i$ and $Q_i$ is the flow rate through the component of subscript $i$ .

The damping effect in the semi-active fluid bypass inerter arises from the pressure drop across the hydraulic cylinder’s chamber $\mathrm{\Delta }p$ . As the helical channel is connected in parallel with the bypass channel, they share the same pressure drop. Therefore, to calculate the damping force of the device, one can focus on the properties and dimensions of the helical channel alone. In their research on fluid inerters, Swift et al. (2013) determined that the dominant source of damping effect originates from the pressure drop resulting from fluid dynamics within the helical channel, including its inlet and outlet. By adapting their mathematical model and incorporating the flow rate ratio $Q_h/Q_1$ , the damping coefficient of the VIBFI can be derived as:

$$\begin{array}{c}\multicolumn{1}{c}{c=0.03426\frac{2}{\sqrt{2}}\left(\frac{\rho l_H}{A_H^2\sqrt{r_H{r}_{\mathrm{cH}}}}\right){\left(\frac{Q_H}{Q_1}\right)}^2{A}_1^3|\stackrel{·}{x}(t)|}\\ +17.54\frac{\mu l_H}{2A_H{r}_H^2}\left(\frac{Q_H}{Q_1}\right)A_1^2+0.75\rho {\left(\frac{A_1}{A_H}\right)}^2|\stackrel{·}{x}(t)|\hfill \end{array}$$

(3)

where $\mu$ is the viscosity of the working fluid, $\stackrel{·}{x}$ is the velocity of the piston, $r_i$ is the hydraulic radius of channel $i$ and $r_{\mathrm{ci}}$ is the radius of curvature of the channel denoted by subscript $i$ . The mathematical models of the VIBFI suggests that the damping coefficient of the device is also affected by $Q_h/Q_1$ .

2.3. Design and prototype

The experimental prototype of the variable inertance bypass fluid inerter was designed and fabricated for this work. Figure 4 presents the CAD model of the variable inertance bypass fluid inerter, while Figure 5 presents the experimental prototype.

Figure 4.

CAD model of the variable inertance bypass fluid inerter.

Figure 5.

Experimental prototype of variable inertance bypass fluid inerter. Subfigures (a) and (b) depicts different views of the prototype.

The prototype was constructed using a combination of off-the-shelf and custom-made components. An off-the-shelf double-rod, double-acting cylinder was employed. A custom-made mount bracket was assembled to the cylinder. Wire-reinforced clear tubing was selected for constructing the helical and bypass channels due to its ease of forming a helical coil, while also maintaining the necessary strength and rigidity. This clear tubing also facilitated the detection of air bubbles in the hydraulic system, which should be avoided in practical applications. The helical channel was created by coiling this tubing around a rigid aluminium cylinder. For plumbing the system, standard threaded connectors were used. A hall effect flow sensor (model: YF-B7) was integrated into the bypass channel to monitor the flow rate, from which the inerter’s flow rate ratio was derived. A manually operated ball valve was used to adjust the flow rate across the bypass channel, thereby controlling the device’s inertance.

According to the mathematical models of the VIBFI, the ideal working fluid should possess high density and low viscosity to prioritise inertial forces over damping forces. Using high viscosity fluids would enhance viscous damping, hindering the desired inertial effect of the device. Additionally, low compressibility is crucial to avoid introducing an elastic-like effect akin to connecting a low-stiffness spring in series with the VIBFI. Water was employed as the working fluid for its availability, cleanliness, incompressibility and low viscosity. A method was adopted to fill the fluid inerter system with water that minimised air entrapment, ensuring optimal performance of the hydraulic system.

Table 1 outlines the design parameters of the device, including primary dimensions and fluid properties.

Table 1.

Prototype design parameters.

Parameter	Symbol	Value
Piston diameter	$d_1$	$0.1m$
Rod diameter	$d_r$	$0.025m$
Bypass channel inner radius	$r_B$	$0.005m$
Helical channel inner radius	$r_H$	$0.005m$
Helical channel curvature radius	$r_{\mathrm{cH}}$	$0.0575m$
Bypass channel curvature radius	$r_{\mathrm{cB}}$	$0.190m$
Helical channel length	$l_H$	$4.460m$
Bypass channel length	$l_B$	$0.480m$
Working fluid type		Water
Working fluid density	$\rho$	$1000\mathrm{kg}{m}^{-3}$
Working fluid viscosity (at $20°C$ )	$\mu$	${10}^{-3}\mathrm{Pa}.s$
Magnitude of dry friction	$f_s$	$80N$
Prototype gross mass	$m$	$10\mathrm{kg}$

3. Experimental and simulation methodology

3.1. Overview

Experimental and simulation studies were conducted to evaluate the VIBFI’s performance and validate its mathematical models. The prototype was tested under steady-state sinusoidal displacement excitation at its terminals, of which the displacement $x$ is represented as:

$$x=x_{\max }\sin (2\pi \mathrm{ft})$$

(4)

with $x_{\max }$ being the excitation amplitude, $f$ is the excitation frequency and $t$ is the elapsed time. The tests aimed to study the impact of adjusting the flow rate through the bypass channel, excitation amplitude and frequency on the variable inertance’s performance parameters, including inertance and damping coefficient. The experiments were performed using a computer-controlled MTS testing platform (Load Frame Model: 370.02, MTS Systems Corp) with built-in force and displacement sensors, as shown by Figure 6. Sinusoidal excitations were applied to the inerter’s terminal by the platform’s hydraulic actuators. Acquisition of the inerter’s terminal force and displacement data as well as control of the excitation were performed using the MTS platform’s built-in systems and software. A separate NI-myRIO module, monitored by another computer, was implemented for the data acquisition from the flow sensor.

Figure 6.

Experimental setup of semi-active fluid inerter.

Simulations were conducted using MATLAB to validate the mathematical models of the VIBFI. The models of the inerter were implemented into the MATLAB program, subjecting to the sinusoidal excitations and independent variables identical to those used in the experiments. Experimental and simulation results were then compared against each other. The mathematical model of the device is validated if experimental results align with simulations closely.

3.2. Experimental procedures

The inerter prototype was installed to the MTS system such that its first terminal was fixed, while its second terminal was subjected to sinusoidal excitation. The MTS system was programmed to exert sinusoidal displacement excitation such that the amplitude and frequency were adjustable by the operator. To investigate the effect of different oscillation frequencies on the prototype’s performance parameters, three frequencies − 0.4, 0.8 and 1.2 Hz were tested. For each frequency, the amplitude was maintained at 5.0 mm. At each excitation condition, 10 different flow rate ratios $Q_h/Q_1$ were tested by adjusting the flow control valve to 10 randomly selected distinct valve knob positions. The first valve knob position involved the valve being fully opened, while the 10th valve knob position involved fully closing the valve. The exact value of $Q_h/Q_1$ would be calculated from the flow sensor data and rod velocity after the experiments. Similarly, to study the effect of the excitation amplitude on the performance parameters, the frequency was kept at 0.8 Hz while three amplitudes − 2.5, 5.0 and 7.5 mm were tested. The identical valve knob positions to the frequency experiments were used for amplitude experiments. The flow chart in Figure 7 shows the process of controlling the independent variables (frequency, amplitude and valve knob position) throughout the experiment.

Figure 7.

Flow chart showing the process of controlling the independent variables in the experiment.

Each change in the independent variables is referred to as a ‘data point’. At each data point, the force-displacement response ellipse of the inerter was recorded as its terminals undergo sinusoidal displacement excitation. Data from these force-displacement response ellipses were then being processed to evaluate the performance of the VIBFI at each data point. Simultaneously, the flow sensor records the volume of fluid that flows through the bypass channel. This volume data was then processed to determine the flow rate ratio $Q_h/Q_1$ of the data point. Detailed data processing procedures implemented by this study are outlined in section 3.4.

3.3. Simulation procedures

Simulations were conducted by subjecting the mathematical models of the VIBFI to the sinusoidal displacement excitation, velocity and acceleration. The parameters $x(t)$ , $\stackrel{·}{x}(t)$ and $\stackrel{··}{x}(t)$ of equation (1) were substituted by:

$$x\left(t\right)=x_{\max }\sin (2\pi \mathrm{ft})$$

(5)

$$\stackrel{·}{x}\left(t\right)=2\pi f{x}_{\max }\cos (2\pi \mathrm{ft})$$

(6)

$$\stackrel{··}{x}\left(t\right)=-{\left(2\pi f{x}_{\max }\right)}^2\sin (2\pi \mathrm{ft})$$

(7)

where $x(t)$ , $\stackrel{·}{x}(t)$ and $\stackrel{··}{x}(t)$ are the excitation displacement, velocity and acceleration respectively. Substituting equations (5)–(7) into (1), the output force can be calculated by:

$$\begin{array}{c}\multicolumn{1}{c}{F\left(t\right)=-b{\left(2\pi f{x}_{\max }\right)}^2\sin \left(2\pi \mathrm{ft}\right)+c2\pi f{x}_{\max }\cos \left(2\pi \mathrm{ft}\right)}\\ +f_s\mathrm{sign}\left(2\pi f{x}_{\max }\cos \left(2\pi \mathrm{ft}\right)\right)\hfill \end{array}$$

(8)

The values of $b$ and $c$ can be calculated using equations (2) and (3), where an array of evenly spaced values from 0.1 to 1 was implemented for $Q_h/Q_1$ . To calculate $c$ , the instantaneous value of $|\stackrel{·}{x}|$ determined equation (6) was also substituted to equation (3). From the vectors of $F(t)$ and $x(t)$ , the simulated force-displacement response ellipses of the VIBFI were plotted. From these ellipses, the simulated performance parameters of the inerter can be determined using data processing procedures.

3.4. Data processing procedures

3.4.1. Flow rate ratio

Data processing procedures were developed to determine the flow rate ratio and performance parameters (inertance and equivalent damping coefficient) of the VIBFI at each data point. In this work, the flow rate ratio $Q_h/Q_1$ was determined with the assumption that the inerter was excited by an ideal sinusoidal displacement during experiments. Because sinusoidal displacement excitation was used to test the inerter, $Q_1$ can be determined by:

$$Q_1=2\pi f{x}_{\max }{A}_1\cos \left(2\pi \mathrm{ft}\right)$$

(9)

where $A_1=(\pi (d_1^2-d_r^2))/4$ is the cylinder’s effective area and $t$ is the elapsed time. Because $Q_1$ is a cosine function, its maximum value is $Q_{1,\max }=2\pi f{x}_{\max }{A}_1$ . The flow sensor records the time series fluid volume that flows through the bypass channel. From there, the volume data was smoothed using the Adjacent-Averaging method and then numerically differentiated to obtain the flow rate. Since this flow rate is also sinusoidal, the maximum flow rate through the bypass channel $Q_{b,\max }$ was used to calculate $Q_h/Q_1$ . From this data, $Q_h/Q_1$ can be determined by:

$$\frac{Q_h}{Q_1}=1-\frac{Q_{b,\max }}{2\pi f{x}_{\max }{A}_1}$$

(10)

where $Q_{b,\max }$ is provided by the flow sensor at that data point. The performance parameters determined were then plotted against $Q_h/Q_1$ for interpretation of the results.

3.4.2. Inertance

The inertance $b$ was calculated by first determining the gradient ∇ of the force-displacement response ellipse, and then dividing the result by negative squared of the oscillation’s angular velocity $-{\omega }^2$ . This approach is equivalent to dividing the equivalent stiffness $K_{\mathrm{eff}}$ by $-{\omega }^2$ , as shown in equation (11):

$$b=-\frac{\nabla }{{\omega }^2}=-\frac{K_{\mathrm{eff}}}{{\omega }^2}$$

(11)

To determine the bisector line for inertance calculation, one can compute the average value of terminal forces exerted by the inerter at the same terminal displacement level in its cycle across a range of terminal displacements. This range spans from the negative to the positive nominal amplitudes of the excitation. The linear equation of this bisector line is approximated using the Least Square method, which facilitates the determination of its gradient. Figure 8 illustrates an example of this bisector line, where the experimental data of the VIBFI under 5 mm sinusoidal excitation was used for illustration.

Figure 8.

Force-displacement response graph with bisector line to illustrate the calculation process of inertance. The force values on the bisector line were calculated by taking the average of the force value on the top curve and bottom curve at the same terminal displacement level.

3.4.3. Damping coefficient

Under sinusoidal excitation, the equivalent damping coefficient $c_{\mathrm{eq}}$ of the device at each data point was calculated by dividing energy dissipated by damping effect $W_{\mathrm{dam}}$ by a function of frequency $f$ and amplitude $x_{\max }$ (Li et al., 2013). $W_{\mathrm{dam}}$ can be determined from experimental or simulation data by numerically calculating the area enclosed by the force-displacement response ellipse. From there, the equivalent damping coefficient $c_{\mathrm{eq}}$ can be calculated by:

$$c_{\mathrm{eq}}=\frac{W_{\mathrm{dam}}}{2{\pi }^{2}f{x_{\max }}^2}$$

(12)

3.4.4. Mathematical model validation

The experimental and simulated performance parameters (inertance and equivalent damping coefficient) are compared to validate the mathematical models of the VIBFI. The percentage error approach was implemented for validation of the mathematical model from experimental results. The percentage error $E$ between the experimentally determined performance parameter $P_{\exp }$ and the simulated performance parameter $P_{\mathrm{sim}}$ at each data point can be determined by equation (13).

$$E=\left|\frac{P_{\exp }-P_{\mathrm{sim}}}{P_{\mathrm{sim}}}\right|\times 100\mathrm{\%}$$

Subsequently, the average percentage error $\overline{E}$ across 10 values of $Q_h/Q_1$ can be calculated to represent the average percentage error between the simulated and experimental performance parameter at a particular excitation condition. A low value of this error would validate the mathematical model of the VIBFI, as it indicates close proximity between experimental and simulated results.

4. Results

4.1. Experimental results

4.1.1. Controllability of $Q_h/Q_1$

The relationship between valve knob position and flow rate ratio $Q_h/Q_1$ of the VIBFI was investigated to assess the controllability of this ratio by modulating the bypass channel’s restriction. During the experiments, the flow control valve was closed gradually from valve knob position 1–7, then partially opened at position 8 and 9 before fully closing at position 10. Figure 9(a) plots $Q_h/Q_1$ at different valve positions with constant excitation frequency and varying amplitude, while Figure 9(b) shows the relationship under constant frequency and varied amplitudes. Both figures demonstrate that adjusting the flow control valve effectively modulates $Q_h/Q_1$ , with values of increasing when the flow control valve gradually closes, and decreasing as it opens. The values of $Q_h/Q_1$ ranges from 0.18 when the flow control valve is fully opened, to 1.0 when the flow control valve is fully closed. In Figure 9(a), $Q_h/Q_1$ remains fairly consistent at the same valve knob position across frequencies, with a slight decrease as the frequency increases. At the seventh valve knob position, $Q_h/Q_1$ reaches unity at 0.4 Hz due to nearly closed valve condition and the flow of fluid is below the measurement range of the sensor. Figure 9(b) indicates a slight increase in $Q_h/Q_1$ at the same valve position at higher excitation amplitudes. This effect is more pronounced with more opened valve and diminishes as the valve closes.

Figure 9.

Plotting the flow rate ratio $Q_h/Q_1$ at different valve knob position. Subfigure (a) plots the flow rate ratio at tested valve position for variable frequency test, while subfigure (b) plots that for variable amplitude test.

4.1.2. Force-displacement response ellipses

Figure 10 plots the force-displacement response ellipses of the experimental prototype under a steady state sinusoidal excitation response of 5 mm amplitude and 0.8 Hz frequency. In the figure, each ellipse presents the force-displacement response of the VIBFI at a specific flow rate ratio $Q_h/Q_1$ , as indicated by the legends. It is seen that increasing $Q_h/Q_1$ increases the negative angle between the major axis of the response ellipse and the horizontal axis. Additionally, increasing $Q_h/Q_1$ also increases the area enclosed by the ellipse. These results suggested that under a constant excitation condition, increasing $Q_h/Q_1$ increases the negative stiffness and energy dissipation of the VIBFI. This trend held true across all tested sinusoidal excitation amplitudes and frequencies.

Figure 10.

Force versus displacement response of the SBFI at different flow rate ratio $Q_h/Q_1$ .

To examine how different excitation conditions impact the force-displacement response characteristic of the VIBFI, the responses under different excitation conditions were compared while keeping the valve knob position constant. In Figure 11(a), the force-displacement response ellipses of the VIBFI are plotted for different excitation frequencies at a fixed amplitude of 5 mm. Meanwhile, Figure 11(b) plots the response ellipses at the excitation frequency of 0.8 Hz with varying amplitudes.

Figure 11.

Force versus displacement response of the SBFI at identical flow rate ratio. (a) Depicts the response at different frequencies while (b) depicts the response at different amplitudes.

The observation from Figure 11(a) indicates that as the excitation frequency increases, there is a corresponding increase in the negative angle between the major axis of the response ellipse and the horizontal axis. Simultaneously, there is a rapid expansion in the area surrounded by the ellipse. Conversely, Figure 11(b) illustrates that increasing the sinusoidal excitation amplitude only results in a larger area bounded by the force-displacement response ellipses. These results suggests that while keeping the valve knob position constant, changing the sinusoidal excitation frequency affects both the negative stiffness and energy dissipation of the VIBFI. However, altering the excitation amplitude only affects energy dissipation of the device.

4.1.3. Inertance

The inertance of the VIBFI, as determined from experiments, was plotted against flow rate ratio $Q_h/Q_1$ to assess the impact of this flow rate ratio and sinusoidal excitation conditions on inertance. These results provide insights into the controllability of inertance for this device. In Figure 12(a), the relationship between inertance and flow rate ratio is plotted at varied excitation frequencies, while Figure 12(b) illustrates the same relationship but with varied excitation amplitudes. Both subfigures consistently show that an increase in $Q_h/Q_1$ corresponds to an increase in the inertance of the VIBFI. According to the experimental results, the VIBFI demonstrated its ability to continuously control inertance within the range of 586.96–3036.79 kg by adjusting $Q_h/Q_1$ . This inertance is significantly greater than the gross mass of the prototype, meaning that the VIBFI can exert the inertial force of a significantly larger mass than itself. Different trends in inertance were observed when sinusoidal excitation conditions were altered. Figure 12(a) reveals that increasing the frequency while keeping $Q_h/Q_1$ constant only result in subtle changes in the inertance. In contrast, Figure 12(b) indicates that increasing the amplitude sightly reduces the inertance of the device.

Figure 12.

Relationship between inertance and flow rate ratio $Q_h/Q_1$ , where (a) depicts the relationship at varied frequencies, while (b) depicts the relationship at varied amplitude.

4.1.4. Equivalent damping coefficient

The equivalent damping coefficient of the VIBFI was plotted against $Q_h/Q_1$ under various sinusoidal excitation conditions to assess controllability and the influence of excitation conditions. Figure 13 plots the relationship between the equivalent damping coefficient $c_{\mathrm{eq}}$ of the experimental prototype and the flow rate ratio $Q_h/Q_1$ . In this context, Figure 13(a) portrays the relationship at varying frequencies, while Figure 13(b) illustrate the relationship at varying excitation amplitudes. Both subfigures indicate that an increase in $Q_h/Q_1$ corresponds to an increase in the equivalent damping coefficient of the VIBFI. It was observed that the magnitude of the equivalent damping coefficient of the VIBFI in SI unit is significantly larger than its inertance, topping at $3.5\times {10}^4\mathrm{Ns}{m}^{-1}$ for the most extreme condition tested. Figure 13(a) and (b) both show an increase in the equivalent damping coefficient when the excitation frequency and amplitude are increased while $Q_h/Q_1$ is kept constant. Notably, Figure 13(a) demonstrates a steeper increase in the damping coefficient with rising frequency.

Figure 13.

Relationship between the equivalent damping coefficient and flow rate ratio $Q_h/Q_1$ . (a) Depicts the relationship at variable frequencies, while (b) depicts that at variable oscillation amplitude.

4.2. Comparison between experimental and simulation results

The simulated and experimental results were compared against each other to validate the mathematical models of the VIBFI. Close alignment between experimental and simulated results would indicate validation of the device’s mathematical models. This subsection presents such comparisons regarding the force-displacement response ellipses, inertance and equivalent damping coefficient of the VIBFI.

4.2.1. Force-displacement response

The force-displacement response ellipses were plotted against each other at selected data points for comparison. These data points were chosen to represent the VIBFI’s response when the flow control valve is at its fully closed position ( $Q_h/Q_1=1$ ), across all frequencies and amplitudes to evaluate performance. Figure 14 plots the comparison between experimental and simulated force-displacement response at $Q_h/Q_1=1$ for all excitation conditions. Subfigure (a) to (c) depicts the comparison at the frequencies of 0.4, 0.8 and 1.2 Hz while the amplitude was kept constant at 5 mm; and subfigure (d) to (f) presents the same comparison under excitation amplitudes of 5, 7.5 and 10 mm while the excitation frequency was kept at 0.8 Hz.

Figure 14.

Comparison between the experimental and simulated force-displacement response ellipses of the inerter. Subfigure (a) to (c) depicts the comparison at the frequencies of 0.4, 0.8 and 1.2 Hz while the amplitude was kept constant at 5 mm; while subfigure (d) to (f) presents the comparison under excitation amplitudes of 5, 7.5 and 10 mm while the excitation frequency was kept at 0.8 Hz.

The simulated and experimentally determined force-displacement responses of the VIBFI fairly matches with each other. Perfect alignment was observed in Figure 14(c), where the excitation amplitude and frequency were set at 5 mm and 0.4 Hz respectively. Differences in negative stiffness between simulated and experimental response ellipses were noticeable, but subtle. There are noticeable differences between the areas bounded by the experimentally determined and the simulated response ellipses. Cavitations appear in the geometry of the experimentally determined force-displacement response ellipses. This effect is more pronounced when the amplitude or frequency is high.

4.2.2. Inertance

Figure 15 compares the experimentally determined relationship between inertance and flow rate ratio $Q_h/Q_1$ with the simulated results. Subfigure (a) plots this relationship under varying excitation frequency while subfigure (b) presents the relationship under varied amplitudes. Results presented by Figure 15 demonstrated a fair alignment between experimentally determined inertance and the values obtained from simulating the mathematical models of the VIBFI. At higher values of $Q_h/Q_1$ , better alignment between experimental and simulated results was observed. Both experimentally determined inertance and simulated inertance increases with $Q_h/Q_1$ . However, the relationship between the experimentally determine inertance and flow rate ratio appears to be more linear. The simulated inertance does not change with variation in excitation frequency nor amplitudes when $Q_h/Q_1$ was kept constant. Therefore, curve representing the simulated relationships between $Q_h/Q_1$ and inertance at different excitation conditions have overlapped one another. A similar trend was observed in the experimental results, where there were small but noticeable changes in the inertance when the excitation conditions changed.

Figure 15.

Comparison between experimental and simulated results: Equivalent inertance versus $Q_h/Q_1$ . (a) Presents the relationship at different excitation frequency, whereas (b) presents the relationship at different excitation amplitudes.

Table 2 displays the average percentage error $\overline{E}$ between the experimentally determined inertance and simulated inertance of the VIBFI across the experimented flow rate ratio range at various excitation frequencies and amplitudes. The maximum average error is at 57.95% while the smallest average error is at 36.13%. The figures in Table 2 suggest that the average error between experimentally determined inertance and the simulated inertance tend to decrease as the excitation amplitude or excitation frequency decreases. However, these average errors are relatively high, indicating discrepancies between the experimental results and simulated models.

Table 2.

Average percentage error between the experimentally determined inertance and simulated inertance of the SBFI at different excitation conditions.

Percentage error $E$	Excitation frequency (Constant amplitude, 5 mm)			Excitation amplitude (Constant frequency, 0.8 Hz)
	0.4 Hz	0.8 Hz	1.2 Hz	5 mm	7.5 mm	10 mm
Average error $\overline{E}$	53.95%	57.95%	44.03%	57.95%	45.69%	36.13%

4.2.3. Equivalent damping coefficient

Figure 16 compares the experimentally determined equivalent damping coefficient against simulated results for all tested excitation conditions. Subfigure (a) to (c) depict the comparison under varied frequency and fixed amplitude, while subfigure (d) to (f) show the comparison under varied excitation amplitude with a fixed frequency. To some extent, the experimental data aligns with the simulated results. Discrepancy between experimentally determined and simulated damping coefficient are more noticeable at lower $Q_h/Q_1$ values, while improved alignment between the results is observed as $Q_h/Q_1$ increases. At the frequency of 1.2 Hz, the simulated damping coefficient is consistently lower than the experimentally determined damping coefficients for all $Q_h/Q_1$ values tested. In addition, the experimentally determined relationship between $Q_h/Q_1$ appears to be more linear than the simulated relationship.

Figure 16.

Comparison between experimental and simulated results: Equivalent damping coefficient versus Qh=Q1. Subfigure (a) to (c) depicts the comparison at the frequencies of 0.4, 0.8 and 1.2 Hz while the amplitude was kept constant at 5 mm; while subfigure (d) to (f) presents the comparison under excitation amplitudes of 5, 7.5 and 10 mm while the excitation frequency was kept at 0.8 Hz.

Table 3 presents the average percentage error $\overline{E}$ between the experimentally determined damping coefficient and the simulated damping coefficient of the VIBFI at different excitation conditions. Under the excitation frequency of 0.4 Hz and amplitude of 5 mm, the average error is determined to be 8.30%, indicating a good alignment between experimental and simulated results. However, as the excitation frequency increases, this error tend to rise. The average error is shown to decrease with increasing excitation amplitudes. This suggests that the agreement between the experimental and simulated damping coefficients is better at lower excitation frequency and higher excitation amplitudes. Nevertheless, even with these trends, the error remains relatively high, with the maximum error reaching 63.91%, indicating notable discrepancies between the experimental and simulated results.

Table 3.

Average percentage error between the experimentally determined damping coefficient and simulated damping coefficient of the VIBFI at different excitation frequency and 5 mm amplitude.

Percentage error $E$	Excitation frequency (Constant amplitude, 5 mm)			Excitation amplitude (Constant frequency, 0.8 Hz)
	0.4 Hz	0.8 Hz	1.2 Hz	5 mm	7.5 mm	10 mm
Average error $\overline{E}$	8.30%	34.68%	63.91%	34.68%	28.60%	21.90%

5. Discussion

5.1. Controllability of inertance and damping coefficient

Experimental testing of the VIBFI under sinusoidal excitation was conducted to evaluate the device’s performance. Experimental results shown by Figure 9 revealed that adjusting the knob position of the flow control valve integrated into the bypass channel effectively controls the flow rate ratio $Q_h/Q_1$ of the VIBFI. Figure 10 further demonstrates that increasing $Q_h/Q_1$ increases both the negative stiffness and energy dissipation of the VIBFI. Trends depicted in Figures 12 and 13 aligns with observations from the force-displacement response ellipses of the VIBFI. The findings indicate that increasing the flow rate ratio $Q_h/Q_1$ simultaneously raises the inertance and the equivalent damping coefficient of the VIBFI. Variation in sinusoidal excitation conditions exhibits a subtle impact on inertance, but significantly affect the equivalent damping coefficient. Specifically, the equivalent damping coefficient of the VIBFI increases with higher amplitude or frequency, with a more rapid increase when the frequency increases. The magnitude of this damping coefficient is observed to significantly surpass the inertance of the VIBFI.

The experimental results implies that the VIBFI can provide controllable inertance as well as equivalent damping coefficient. These performance parameters (inertance and damping coefficient) can be continuously adjusted by modulating the flow restriction through the bypass channel, which is achieved by adjusting the opening of the integrated flow control valve. This underscores the feasibility of developing a semi-active inerter with variable inertance and damping coefficient by integrating a parallel bypass channel with a flow control valve into a fluid inerter. The large damping coefficient magnitude and controllability make the VIBFI effective as a device combining a variable inerter and damper. However, for applications that require purely variable inertance, minimising the damping effect of the VIBFI is essential.

5.2. Validation of mathematical model

Simulation results derived from evaluating the mathematical models of the VIBFI under similar independent variables and excitation conditions were compared against experimental results to validate the models. While the mathematical model can predict the real-world behaviour of the VIBFI in some scenario, there were still discrepancies between the model predictions and actual performance of the device. Typically, the model was able to accurately predict the VIBFI’s performance when the flow of fluid is entirely directed to the helical channel (when $\frac{Q_h}{Q_1}=1$ ). Since the operation of the VIBFI is identical to a passive fluid inerter when the bypass channel is blocked, this finding aligns with the mathematical models developed by Swift et al. for passive fluid inerters (Swift et al., 2013). Discrepancies between the experimental and simulated results are observed when $\frac{Q_h}{Q_1}<<1$ . These discrepancies can be attributed to the construction attributes of the experimental prototype.

Firstly, the mathematical model assumes that the internal diameter of the helical channel and the bypass channel remain constant throughout their lengths. These diameters significantly affect the device’s inertance and damping coefficient. While the diameter of the helical channel does remain constant, the bypass channel accommodates components such as fittings, valves and the flow sensor, which often have reduced internal diameter in their construction. This discrepancy explains why the mathematical model can accurately predict the performance parameters when there is no flow through the bypass channel (when $\frac{Q_h}{Q_1}=1$ ), but inaccurately when there is flow through the bypass channel (when $\frac{Q_h}{Q_1}<<1$ ).

Secondly, the experimental prototype utilises a hall-effect type flow sensor that relies on the rotation of an impeller to measure fluid flow. Factors such as friction, efficiency, small misalignments in the assembly and the inertia of the impeller can affect the sensitivity and accuracy of the measurement. This discrepancy can lead to the measured flow rates through the bypass channel differing from the actual flow rates, affecting the measured value of $Q_h/Q_1$ and thus the experimentally determined relationship between the flow rate ratio and the device’s performance parameters.

Thirdly, the vinyl tubing used for the helical and bypass channel, despite being reinforced with steel wire, remains flexible. Consequently, they can expand or contract due to changes in pressure while the terminals of the VIBFI are subjected to relative motion. This flexibility can lead to cavitations in the force-displacement response of the device and unwanted variations in the diameter of the fluid channels (helical and bypass), contributing to the discrepancies between experimental and simulation results. And lastly, despite efforts to minimise air bubbles in the system, there can still be air bubbles trapped in the working volume of the prototype. As air is compressible, this would affect the overall stiffness of the system, causing its actual performance to deviate from that estimated by the mathematical model.

6. Conclusion and future work

This technical paper documented the study of a semi-active inerter prototype dubbed the VIBFI. An experimental prototype of the concept was designed, constructed and tested. Simultaneously, an effort to develop and validate a mathematical model of the VIBFI is thoroughly detailed. The experimental results demonstrate the controllability of performance parameters of the device, including inertance and damping coefficients, through modulating the flow restriction of the bypass channel. The mathematical model derived for the VIBFI can predict its performance effectively when the flow through the bypass channel is blocked. However, there are still discrepancies between the experimental and simulated results. Future work will aim to address these discrepancies to produce a more reliable mathematical model, before integrating active control systems into the device and performing design optimisation.

Future work should focus on improving the prototype of the VIBFI and the mathematical models to achieve better alignment between experimental and simulated results. Improvements in the construction of the prototype are necessary to enhance the agreement of its performance with its mathematical models. Firstly, improvements can be made to the prototype by using more rigid tubing, such as copper or stainless steel, to fabricate the helical channel and the bypass channel. This improvement is expected to reduce the cavitations in the force-displacement response and the unwanted variations in the fluid channel’s diameter, thereby improving the alignment between simulated and experimental data. Secondly, increasing the length of the bypass channel can reduce the effect of inner diameter reductions caused by components installed on it such as flow sensors, valves or fittings. This adjustment ensures that the average inner diameter of the channel closely matches that of the tubing used for the channel. Thirdly, the prototype can also be enhanced by implementing more accurate flow sensors for measuring the flow rate through the bypass channel. The mathematical model also requires refinement to address discrepancies between the simulated and experimental data. Further research is recommended on the dry friction component of the VIBFI, particularly regarding its behaviour under varying excitation conditions. Additionally, the effect of air bubbles trapped in the fluid system on the performance of the VIBFI can be integrated into the mathematical model.

Once the mathematical model and the designs are refined, an active control system can then be integrated to modulate the inertance of the VIBFI. A possible control strategy for the inertance of the VIBFI is as follows: The control system will include an automatic flow control valve where its opening can be controlled based on the desired inertance input and data from the flow sensor as well as the motion of the terminals to achieve the desirable inertance. Measured flow rate from the flow sensor and velocity data of the VIBFI’s terminals can be used to evaluate the current $Q_h/Q_1$ of the device. By using the mathematical model, its measured inertance can be evaluated from the measured value of $Q_h/Q_1$ . The measured inertance is compared against the desirable inertance input, and the opening of the automatic flow control valve can be adjusted by the control system to minimise the difference between the desired and the measured inertance. When there is no motion between the terminals of the VIBFI, the automatic flow control valve can either be fully opened or fully closed, depending on the designer’s preference for their application. In this way, the VIBFI will act as a passive inerter, offering a fail-safe feature for mechanical systems.

The improved mathematical model can be employed to optimise the performance of the VIBFI. Utilising the equations to describe inertance and damping coefficients as objective functions, multiple-objective optimisation methods can be implemented to determine the design dimensions of the VIBFI according to the desired design requirements. For instance, the optimisation method can be used to derive the design dimensions of the VIBFI to maximise inertance while minimising damping as needed. In addition, if the mathematical model of the VIBFI can accurately represent its actual performance, it can be implemented into simulations of semi-active mechanical systems that incorporate the device to provide more realistic results.