Vibration Reduction System with a Linear Motor: Operation Modes, Dynamic Performance, Energy Consumption

Abstract

The study aimed to present the features of a 2-DOF vibration reduction system (VRS) equipped with a linear electrodynamic motor in passive, semi-active and active mode. At first, the VRS model was formulated. Then, simulation tests of the VRS were conducted to distinguish its most advantageous features in each mode and to analyze dynamic performance and energy consumption. Next, the VRS was experimentally tested to evaluate its effectiveness in each mode for the assumed excitations and to compare tests results against simulation data. In active mode, a sliding mode algorithm was employed for motor control, while in semi-active mode, the equivalent damping coefficient analysis was used.

1. Introduction

The division of vibration reduction methods into passive, active and semi-active is based on the possibility of using an additional energy source. Passive methods require that the facility’s operating conditions are defined and appropriate structural changes are made. The effectiveness of these methods is limited by the level of interference not considered by the designer. There is rich literature on this topic, such as from several decades ago, e.g., [1,2,3], and more contemporary literature, e.g., [4,5]. Active methods ensure the effectiveness of activities in the field of excitations limited by the adopted control law and the possibility of supplying energy from an additional source. These methods and their application also have rich literature. Among the older works, we should mention, e.g., [6], among the more recent ones [7,8,9]. On the other hand, semi-active methods use the control law to change system parameters. In contrast to active methods, the additional energy is only used to change the parameters of the actuator employed in the system. The intensive development of semi-active methods over the last six decades has also led to numerous papers, e.g., [10,11,12]. The common point of each method is to consider the dynamics of the facility and the availability of energy sources. For example, in [13,14] a semi-active vibration reduction system with self-powered and self-sensing capability is reported, and in [15] the energy flows in this type of system are studied. When two or more reduction methods are used in one system, it is referred to as a hybrid system or hybrid application, as exemplified by [16,17] for example. Structural dynamics issues are also widely described in the literature. The theoretical basis for this is contained in [18]. Among the numerous practical applications, special attention should be paid to the automotive sector [19,20,21] and the civil engineering sector [22].

Various types of control algorithms are used in vibration reduction systems where active and semi-active methods are applied. In addition to classical PID (proportional integral derivative regulator), skyhook, groundhook, LQR (linear–quadratic regulator), on/off switching algorithms and their modifications, e.g., cascade, optimal, slide mode or adaptive regulators, fuzzy logic or neural network regulators are also used. For example, in [17], the skyhook algorithm was applied for control in the magnetorheological elastomer-based vibration reduction system, in [23,24] the fuzzy logic algorithm for suspension control, and in [25] the negative stiffness algorithm for seat vibration reduction system. Conversely, the use of slide mode algorithms is described in [26,27].

Many researchers focus on an analysis of vibration reduction systems operating in only one mode and do not consider the benefits resulting from the possibility of their combination. In addition, they omit the problem of energy source availability for system power. Therefore, the purpose of the present work is to develop a vibration reduction system (VRS) operating in hybrid mode, taking into account the limitation resulting from the finite capacity of the energy source. To achieve this, we carried out numerical simulations of this system in the passive, semi-active and active modes from the point of view of dynamic performance and energy consumption. Based on the results, the VRS operating in hybrid mode was proposed, and its effectiveness was confirmed experimentally.

The study is structured as follows. Section 2 contains a brief description of the VRS in the test rig and presents governing equations of the system. Section 3 highlights details of the VRS modelling in the passive, semi-active and active modes and presents the sliding mode algorithm employed for motor control. Section 4 discusses the simulation results of the VRS in each mode. Section 5 presents and summarizes data obtained from the carried out experiments. Conclusions are drawn in Section 6. In

Table 1. List of symbols used in the formulas.

Symbol	Description	Symbol	Description
$t$	time	$\sigma$	sliding variable
$t_{max}$	simulation time	$c$	angle of the sliding surface on a state trajectory
$\mathrm{z}$	displacement of shaker plate	$V$	Lyapunov function
$x_1$	displacement of plate 1	$F_{max}$	$\mathrm{maximum} \mathrm{value} \mathrm{of} F_d$
$x_2$	displacement of plate 2	$\beta$	$\mathrm{control} \mathrm{algorithm} \mathrm{parameter} \mathrm{chosen} \mathrm{to} \mathrm{offset} \mathrm{the} \mathrm{impact} \mathrm{of} F_{max}$
$m_1$	mass of plate 1	$A_z$	$\mathrm{amplitude} \mathrm{of} \mathrm{displacement} \mathrm{z}$
$m_2$	mass of plate 2	$A_{x1}$	$\mathrm{amplitude} \mathrm{of} \mathrm{displacement} x_1$
$b_1$	damping coefficient	$A_{x2}$	$\mathrm{amplitude} \mathrm{of} \mathrm{displacement} x_2$
$b_2$	damping coefficient	$f$	frequency of the input signal
$k_1$	spring coefficient	$f_1$	starting frequency
$k_2$	spring coefficient	$f_2$	ending frequency
$F_m$	motor control force	$f_{x1}$	$\mathrm{vibration} \mathrm{frequency} \mathrm{determined} \mathrm{for} x_1$
${\kappa }_1$	force sensitivity coefficient	$K{p}_{x1}$	$\mathrm{overshoot} \mathrm{of} \mathrm{displacement} x_1$
${\kappa }_2$	back emf coefficient	$K{p}_{x2}$	$\mathrm{overshoot} \mathrm{of} \mathrm{displacement} x_2$
$e_m$	electromotive force	$T_{x1\mathrm{z}}$	displacement transmissibility
$R_m$	motor resistance	$T_{x2\mathrm{z}}$	displacement transmissibility
$L_m$	motor inductance	$T_{px1\mathrm{z}}$	$\mathrm{value} \mathrm{of} T_{x1\mathrm{z}}$$\mathrm{in} \mathrm{case} \mathrm{PM} \mathrm{at} \mathrm{the} \mathrm{frequency} f=2 \mathrm{Hz}$
$i_m$	current in motor circuit	$T_{px2\mathrm{z}}$	$\mathrm{value} \mathrm{of} T_{x2z}$$\mathrm{in} \mathrm{case} \mathrm{PM} \mathrm{at} \mathrm{the} \mathrm{frequency} f=2 \mathrm{Hz}$
$u_m$	motor circuit voltage	$D_p$	dynamic performance coefficient
$R_l$	additional resistance	$t_n-t_{n-1}$	$\mathrm{time} \mathrm{duration} \mathrm{of} n$-th displacement cycle
$c_m$	equivalent damping coefficient	$E_{dy}$	dissipated energy in the time duration of one cycle of displacement
$F_c$	control force	$E_{de}$	delivered energy in the time duration of one cycle of displacement
$F_d$	disturbing force	$E_c$	energy consumption coefficient

, we present the designations used in the study.

2. Vibration Reduction System

The 2-DOF VRS investigated in the study is shown in Figure 1 (see dotted line). The first degree of freedom consists of two springs and a movable plate 1. The second degree of freedom is composed of a movable plate 2, two springs and a linear motor acting as an actuator. Plate 1 and plate 2 are mounted on linear guides. The springs making up the first degree of freedom are attached to the shaker plate. The test rig comprises a VRS housed in an aluminum frame, a shaker, a control system and a potentiometer.

The VRS uses an LA25 series electrodynamic linear motor of Sensata Technologies [28]. The motor is composed of two elements (able to move relative to each other), coils system made of resistance wire wound on a diamagnetic core and a ferromagnetic core with an array of permanent neodymium magnets. The VRS extortion is implemented as a shaker plate displacement generated by the LA30 series actuator of Sensata Technologies [29]. S1 and S2 displacement sensors of SMK series [30] are mounted on the shaker plate and plate 1. SMS12 series sensor [31] is mounted on plate 2 (S3). Figure 2 shows a diagram of the VRS. The designations used in the figure mean: $m_1$ and $m_2$—mass of plate 1 and plate 2; $z$, $x_1$, $x_2$—displacement of shaker plate, plate 1 and plate 2; $k_1$ and $k_2$—spring coefficients; $b_1$ and $b_2$—damping coefficients in kinematic pairs; and $F_m$—motor control force. The governing equations of system motion can be written in the form:

$$\left\{\begin{array}{l}{m}_1\ddot{x_1}+b_1\left(\dot{x_1}-\dot{z}\right)+b_2\left(\dot{x_1}-\dot{x_2}\right)+2k_1\left(x_1-\mathrm{z}\right)+2k_2\left(x_1-x_2\right)=-F_m\\ m_2\ddot{x_2}+b_2\left(\dot{x_2}-\dot{x_1}\right)+2k_2\left(x_2-x_1\right)=F_m\end{array}\right.$$

(1)

In view of the fact that the force $F_m$ produced by the motor depends on the force sensitivity ${\kappa }_1$ coefficient and the current $i_m$ in the circuit, and the induced electromotive force $e_m$ is the product of the back emf ${\kappa }_2$ coefficient and the relative velocity $\left(\dot{x_2}-\dot{x_1}\right)$. The constitutive equations of the motor can be written as:

$$\left\{\begin{array}{l}{F}_m={\kappa }_1{i}_m\hfill \\ e_m={\kappa }_2\left(\dot{x_2}-\dot{x_1}\right)\hfill \end{array}\right.$$

(2)

The designations $R_m$ and $L_m$ introduced in Figure 2 indicate the resistance and inductance of the motor coil.

Table 2. Mechanical and electrical parameters of the vibration reduction system.

Mechanical Parameters
$m_1$	$b_1$	$k_1$	$m_2$	$b_2$	$k_2$
$6 \mathrm{kg}$	$10 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$	$2400 \mathrm{N}/\mathrm{m}$	$13.8 \mathrm{kg}$	$13 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$	$2700 \mathrm{N}/\mathrm{m}$
Electrical Parameters
$R_m$	$L_m$	${\kappa }_1$		${\kappa }_2$
$2.4 \mathsf{\Omega }$	$2.5 \mathrm{mH}$	$21.35 \mathrm{N}/\mathrm{A}$		$21.36 \mathrm{V}\cdot \mathrm{s}/\mathrm{m}$

provides the experimentally determined values of the test rig’s mechanical and electrical parameters (see Figure 1).

3. Modelling

The structure of the motor allows the VRS to operate in passive, semi-active or active mode (Figure 3). In passive mode, the model of the motor is achieved by including a fixed resistance $R_l$ in the circuit. In semi-active mode, a controllable load $R_l$ should be included in the motor circuit, and in active mode a controllable voltage or current amplification circuit.

3.1. Passive and Semi-Active Mode

Passive and semi-active VRS mode require the inclusion of additional resistance $R_l$ in the motor circuit. According to Faraday’s law of electromagnetic induction, an electromotive force $e_m$ is induced in the motor circuit proportional to the relative velocity $\left(\dot{x_2}-\dot{x_1}\right)$. The governing equation of the motor circuit is as follows:

$${\kappa }_2\left(\dot{x_1}-\dot{x_2}\right)=L_m\dot{i_m}+(R_m+R_l)i_m$$

(3)

Taking into account that at low frequencies of $z$, the inductance $L_m$ takes on small values, the force generated by the motor can be written as:

$$F_m\approx c_m\left(\dot{x_1}-\dot{x_2}\right)$$

(4)

where $c_m$ is an equivalent damping coefficient:

$$c_m=\frac{{\kappa }_1{\kappa }_2}{R_m+R_l}$$

(5)

Equation (4) shows that the motor influence on the VRS is viscous so that the generated force depends on the damping coefficient $c_m$. The value of $c_m$ can be modified by changing the value of the resistance $R_l$ according to the graph shown in Figure 4.

3.2. Active Mode

In active mode, a source of voltage $u_m$ or current $i_m$ must be included in the linear motor circuit. Then, the equation of the motor circuit takes the form:

$$u_m=L_m\dot{i_m}+R_m{i}_m+e_m$$

(6)

The purpose of the active mode is to minimize the amplitude of vibration in the system. To achieve this, a sliding mode control algorithm was chosen. The equation of plate 1 motion can be written as:

$$\ddot{x_1}=\frac{1}{m_1}{F}_d\left(x_1,\dot{x_1},x_2,\dot{x_2},z,\dot{z},t\right)+\frac{1}{m_1}{F}_c\left(i_m,t\right)$$

(7)

where $F_c$ is the control force that should counteract the force $F_d$ disturbing the accepted equilibrium point ($x_1=0$ and $\dot{x_1}=0$). Considering Equation (2), the force $F_c$ and $F_d$ can be expressed as:

$$F_c\left(i_m,t\right)=-{\kappa }_1{i}_m$$

(8)

$$F_d\left(x_1,\dot{x_1},x_2,\dot{x_2},z,\dot{z},t\right)=b_1\left(\dot{z}-\dot{x_1}\right)+b_2\left(\dot{x_2}-\dot{x_1}\right)+2k_1\left(z-x_1\right)+2k_2\left(x_2-x_1\right)$$

(9)

Let us assume the sliding variable $\sigma$ in the form:

$$\sigma =\dot{x_1}+c{x}_1$$

(10)

where $c$ is the angle of the sliding surface on a state trajectory. To formulate the control law, it is necessary to take the Lyapunov function $V$, which is a measure of the distance of the state trajectory from the sliding surface [32]:

$$V=\frac{1}{2}{\sigma }^2$$

(11)

To ensure asymptotic stability at the equilibrium point $\sigma =0$, the function $V$ must satisfy the following conditions [33]:

$$\underset{\left|\sigma \right|\to \infty }{\lim }V=\infty$$

(12)

$$\dot{V}<0 \mathrm{for} \sigma \ne 0$$

(13)

Taking into account Equations (7), (8), (10) and (11), the derivative of the function $V$ can be expressed as follows:

$$\dot{V}=\sigma \dot{\sigma }=\sigma \left(\ddot{x_1}+c\dot{x_1}\right)=\sigma \left(\frac{1}{m_1}{F}_d\left(x_1,\dot{x_1},x_2,\dot{x_2},z,\dot{z},t\right)-\frac{{\kappa }_1}{m_1}{i}_m+c\dot{x_1}\right)$$

(14)

If the control signal is assumed to be:

$$i_m=\frac{m_1}{{\kappa }_1}c\dot{x_1}+\frac{1}{{\kappa }_1}\beta ·\mathrm{sign}\left(\sigma \right)$$

(15)

Then, satisfying condition (13) requires constraining the maximum value of $\dot{V}$ according to the inequality:

$$\sigma \left(\frac{1}{m_1}{F}_d-\frac{1}{m_1}\beta ·\mathrm{sign}\left(\sigma \right)\right)<\left|\sigma \right|\left(\frac{1}{m}{F}_{max}-\frac{1}{m_1}\beta ·\mathrm{sign}\left(\sigma \right)\right)$$

(16)

where $F_{max}$ is the maximum value of $F_d$, and $\beta$ is a parameter chosen to offset the impact of $F_{max}$. Note that inequality (16) will be satisfied when $\beta \ge F_{max}$.

The use of a control signal as in Equation (15) requires the introduction of an additional current control system to compensate current changes due to the induced electromotive force $e_m$. Assuming the energy source is a voltage generating system $u_m$ and ignoring the small influence of the inductance $L_m$, two components can be distinguished in the force $F_m$ resulting from the interaction of the external voltage source and changes in relative velocity $\left(\dot{x_2}-\dot{x_1}\right)$:

$$F_m\approx \frac{{\kappa }_1}{R_m}{u}_m+\frac{{\kappa }_1{\kappa }_2}{R_m}\left(\dot{x_1}-\dot{x_2}\right)$$

(17)

The component $\frac{{\kappa }_1{\kappa }_2}{R_m}\left(\dot{x_1}-\dot{x_2}\right)$ can be treated as an additional component of the disturbing force $F_d$. In connection with the control force $F_c$ can be written as:

$$F_c\left(u_m,t\right)=-\frac{{\kappa }_1}{R_m}{u}_m$$

(18)

The adopted modification of the force $f_c$ allows the assumption of the control signal $u_m$ in the form:

$$u_m=\frac{m_1{R}_m}{{\kappa }_1}c\dot{x_1}+\frac{R_m}{{\kappa }_1}\beta ·\mathrm{sign}\left(\sigma \right)$$

(19)

4. Simulation Tests

Numerical simulations of the VRS in each mode (see Figure 3) were carried out using Equation (2). In passive mode (case PM), Equation (3) was additionally used, assuming $R_l=200 \mathsf{\Omega }$. In the semi-active mode, Equation (3) was used, assuming that the control system can modify the value of the resistance $R_l$ in the range from $0$ to $25 \mathsf{\Omega }$. Exemplary simulations were performed with $R_l=0 \mathsf{\Omega }$ (case SM1), $R_l=2 \mathsf{\Omega }$(case SM2), $R_l=5 \mathsf{\Omega }$(case SM3), and $R_l=10 \mathsf{\Omega }$ (case SM4). In the active mode (case AM), Equation (6) and the control law written by Equation (19) were used, assuming from work [34] that $\beta =50 \mathrm{N}$ and $c=10 {\mathrm{s}}^{-1}$. Simulations were performed with a step signal:

$$z\left(t\right)=\left\{\begin{array}{c}0 \mathrm{m},\\ 0.1 \mathrm{m},\end{array} \right.\begin{array}{c}\mathrm{for} t<0\\ \mathrm{for} t\ge 0\end{array}$$

(20)

Additionally, a sinusoidally varying displacement with amplitude $A_z=0.01 \mathrm{m}$ and slowly increasing frequency from $f_1=0.1$to $f_2=10 \mathrm{Hz}$ in $t_{max}=300 \mathrm{s}$:

$$z\left(t\right)=A_z\sin \left[2\mathsf{\pi }t\left(f_1+t\frac{f_2-f_1}{2t_{max}}\right)\right]$$

(21)

To evaluate the effectiveness of VRS under step signal [35], the following overshoot of displacement $x_1$ and $x_2$ were introduced:

$$K{p}_{x1}=\frac{\max \left(x_1\right)-0.1}{0.1}·100\%$$

$$K{p}_{x2}=\frac{\max \left(x_2\right)-0.1}{0.1}·100\%$$

(22)

For sinusoidally varying excitation, displacement transmissibility was defined as:

$$T_{x1\mathrm{z}}\left(f\right)=\frac{A_{x1}}{A_z}$$

$$T_{x2\mathrm{z}}\left(f\right)=\frac{A_{x2}}{A_z}$$

(23)

where the coefficients $A_{x1}$ and $A_{x2}$ denote the displacement amplitudes $x_1$ and $x_2$. Finally, it was assumed that the dynamic performance coefficient $D_p$ would determine effectiveness of the VRS:

$$D_p=100-\frac{1}{4}K{p}_{x1}-\frac{1}{4}K{p}_{x2}-\frac{T_{x1\mathrm{z}}\left(2\right)}{T_{px1\mathrm{z}}}·25\%-\frac{T_{x2z}\left(2\right)}{T_{px2\mathrm{z}}}·25\%$$

(24)

where $D_p$ is expressed as a percentage, $T_{x1\mathrm{z}}\left(2\right)$ and +$T_{x2z}\left(2\right)$ refer to the frequency $f=2 \mathrm{Hz}$, and $T_{px1\mathrm{z}}$ and $T_{px2\mathrm{z}}$ specify the value of $T_{x1\mathrm{z}}$ and $T_{x2z}$ in case PM at the frequency $f=2 \mathrm{Hz}$. The dynamic performance coefficient $D_p$ allows the evaluation of the performance of the VRS for frequencies below or close to the first resonant frequency of the system.

Assuming the existence of an external energy source in VRS in the case AM, the development of the energy consumption coefficient requires separate consideration of the cases PM, SM1, SM2, SM3, and SM4, in which there is no external power source. In PM, SM1-SM4 cases, the mechanical energy provided by the shaker is converted to electrical energy and dissipated at the resistances $R_m$ and $R_l$. The amount of energy $E_{dy}$ that can be generated and stored in the time duration of one cycle of displacement $z$ is equal to the energy dissipated on the resistance $R_l$:

$$E_{dy}=-{{\displaystyle \int }}_{tn-1}^{tn}{R}_l·i_m{}^2·\mathrm{d}t$$

(25)

where $t_n-t_{n-1}$is the time duration, and $n$ is the cycle number.

If an external energy source is connected, the amount of delivered energy $E_{de}$ can be defined as:

$$E_{de}={{\displaystyle \int }}_{tn-1}^{tn}\frac{u_m{}^2}{R_m}·\mathrm{d}t$$

(26)

Considering Equations (24) and (26), the energy consumption coefficient is given by:

$$E_c=\left\{\begin{array}{l}{E}_{dy}\left(2.15\right), \mathrm{case} \mathrm{PM}, \mathrm{SM}1–\mathrm{SM}4\\ E_{de}\left(2.15\right) , \mathrm{case} \mathrm{AM} \hfill \end{array}\right.$$

(27)

where $E_{dy}\left(2.15\right)$ and $E_{de}\left(2.15\right)$ refer to the frequency $f=2.15 \mathrm{Hz}$. The frequency value was selected based on the $E_{dy}\left(f\right)$ local maximum value.

Simulation results under such assumptions are shown in Figure 5, Figure 6 and Figure 7. The plots in Figure 5b,c shows the VRS response to the step signal (Figure 5a) in the form of a time waveform $x_1$ and $x_2$ displacements. It can be noted that the smallest overshoot value occurs in AM case and the largest in PM case. Moreover, when analysing cases SM1–SM4, it can be seen that increasing the value of $R_l$ causes an increase in the value of overshoot.

The simulation results of the VRS under sinusoidally varying excitations are shown in Figure 6a,b. The plots show displacement transmissibility $T_{x1\mathrm{z}}$ and $T_{x2z}$. For greater readability of the graphs, cases SM3 and PM have been omitted. Considering transmissibilities $T_{x1\mathrm{z}}$ and $T_{x2z}$ at low frequencies $f$, the most favourable case for VRS operation is the AM case. In other cases, effectiveness depends on frequency. In the range from $0.1$ to $2.3 \mathrm{Hz}$ and from $5.7$ to $10 \mathrm{Hz}$, a small value of $R_l$ improves the efficiency of the VRS. However, between $2.3$ and $5.7 \mathrm{Hz}$, a small value of $R_l$ reduces the effectiveness of the VRS.

Figure 7 shows the relationship of the dissipated energy $E_{dy}$ and the energy supplied $E_{de}$ from an external power source under sinusoidally varying excitations. It can be seen that in the case AM, there is high energy consumption. In case PM and SM2–SM4, the amount of energy $E_{dy}$ depends on the frequency $f$ and the resistance $R_l$. Analysing the influence of the resistance $R_l$ on the energy $E_{dy}$, it can be noticed that at each of the frequency values adopted for the simulation, there is a particular value $R_l$, for which $E_{dy}$ takes on a maximum value. For example, Figure 8 presents the dependence of the dissipated energy $E_{dy}$ on the resistance $R_l$ at frequency $f=2.1 \mathrm{Hz}$.

In

Table 3. Factors of the vibration reduction system in the passive, semi-active and active mode.

Mode	$\mathit{K}{\mathit{p}}_{\mathit{x}1}$ $\left[\mathit{\%}\right]$	$\mathit{K}{\mathit{p}}_{\mathit{x}2}$ $\left[\mathit{\%}\right]$	${\mathit{T}}_{\mathit{x}1}\left(2 \mathbf{Hz}\right)$ $\left[-\right]$	${\mathit{T}}_{\mathit{x}2}\left(2 \mathbf{Hz}\right)$ $[-]$	${\mathit{D}}_{\mathit{p}}$ $\left[\mathit{\%}\right]$	${\mathit{E}}_{\mathit{c}}$ $\left[\mathbf{J}\right]$
PM	79.3	103.7	19.69	33.55	4.25	−0.67
SM1	56.8	76.3	2.02	2.94	61.97	0
SM2	59.2	77.2	2.97	4.74	58.60	−0.74
SM3	66.5	86.5	4.36	7.23	50.83	−1.49
SM4	71.6	93.1	6.35	10.7	42.79	−2.14
AM	43.5	45.1	0.75	0.70	76.38	8.11

we show the factors of VRS in each considered mode. In addition to the previously introduced parameters $K{p}_{x1}, K{p}_{x2}$, $T_{x1\mathrm{z}}$, and $T_{x2z}$ the dynamic performance coefficient $D_p$ and energy consumption coefficient $E_c$ are also given. The $D_p$ coefficient confirms the VRS effectiveness in case AM. Considering the other cases, the value of the coefficient $D_p$ depends on the $R_l$ resistance and decreases with its increase. Considering the $E_c$ coefficient, it should be noted that the amount of recovered energy can be optimized by matching the resistance $R_l$ to the frequency $f$ of the sinusoidal excitation. In case AM, the amount of energy supplied from an external source is at least two times greater than the amount of energy recovered in each of the analyzed cases. The obtained results make it possible to define the principles of VRS operating in hybrid mode. Within the adopted range of frequency $f$, five ranges can be distinguished that define the operation of the VRS in a particular mode:

• Up to $1.9 \mathrm{Hz}$, SM at $R_l=11 \mathsf{\Omega }$, increasing the amount of recovered energy;
• From $1.9$ to $2.7 \mathrm{Hz}$, AM in the case of available energy from an external source or SM at $R_l=0 \mathsf{\Omega }$ when no external source is available, reducing the values of $T_{x1\mathrm{z}}$ and $T_{x2\mathrm{z}}$;
• From $2.7$ to $5.7 \mathrm{Hz}$, SM at $R_l=11 \mathsf{\Omega }$, $T_{x1\mathrm{z}}$ value reduction and increase the amount of recovered energy;
• From $5.7$ to $9.0 \mathrm{Hz}$, SM at $R_l=0 \mathsf{\Omega }$, value reduction of $T_{x1\mathrm{z}}$;
• Ranging from $9.0 \mathrm{Hz}$, SM at $R_l=2 \mathsf{\Omega }$, value reduction of $T_{x1\mathrm{z}}$, increasing the amount of recovered energy.

5. Experimental Tests

A block diagram of the control system designed for the test rig is shown in Figure 9. The control system is composed of an industrial cRIO 9063 controller [36], a potentiometer, a PWM module, and a PC. The cRIO controller includes an FPGA module, RT processor and chassis designed for measurement cards. Communication between the RT processor and the FPGA module is via DMA bus. On the other hand, the FPGA module communicates with the measurement cards and the PWM module by I/O bus. The chosen architecture enables the implementation of control algorithms hardware-based on the FPGA module and software-based in the RT-Linux operating system.

Measurement cards were configured to allow the measurement of displacement $z$, $x_1$, $x_2$ voltage $u_m$, and current $i_m$. The sampling rate of the acquired data archived on the solid-state drive was $4 \mathrm{kHz}$. The defined process data and settings were verified in a control panel application running on an independent PC. The controller software and the control panel application were developed in the LabView environment.

The linear motor can operate in active mode by drawing energy from an external source or in semi-active mode by dissipating the generated energy in a resistive load circuit. In active mode, the power conditioner is a commercial NI 9505 PWM [37] module. In semi-active mode, the resistive nature of the load was achieved using a potentiometer $R_l$ (operating in range from $1$ to $25 \mathsf{\Omega }$). When working in active mode, the physical limitations of the linear motor must be taken into account. The instantaneous value of the current $i_m$ is limited by the amount of heat energy that must be dissipated by the cooling system. In practice, assuming motor operation at room temperature and passive air cooling, the motor can be loaded momentarily with $9 \mathrm{A}$ or continuously with $4 \mathrm{A}$. Therefore, the continuous force should not exceed $80 \mathrm{N}$, and the voltage $u_m$ should be between $-9$ and $9 \mathrm{V}$.

The control algorithm implementation in VRS is divided into three steps: data acquisition, frequency determination, and frequency range selector (see Figure 10).

In step 1, implemented in the FPGA module, the acquired data were subjected to conditioning and filtering the displacement $x_1$ and velocity $\dot{x_1}$. The displacement $x_1$ was sent to the RT system using a DMA buffer. Step 2 was implemented in the RT system in a separate thread sampled at a frequency of $100 \mathrm{Hz}$. In each processing cycle, the RT system updated a buffer containing the $8192$ most recently received displacement $x_1$ samples and determined the dominant frequency $f_{x1}$. In step 3, the frequency $f_{x1}$ was sent to the FPGA module (using a shared memory buffer) to calculate the values of $\beta$, $c$ and $R_l$. Assuming VRS in semi-active and hybrid mode, the frequency range selector determined the coefficients $\beta$, $c$ and $R_l$ following the frequency ranges provided in

Table 4. Parameters of the control algorithm; semi-active and hybrid mode.

Semi-Active Mode
$\mathbf{Frequency} \mathbf{Range} \mathbf{Hz}$		$\mathit{\beta } N$	$\mathit{c} s^{-1}$	${\mathit{R}}_{\mathit{l}} \Omega$
I	$f_{x1}\le 2.1$	$0$	$0$	$1$
II	$2.1<f_{x1}\le 4.1$	$0$	$0$	$25$
III	$4.1<f_{x1}$	$0$	$0$	$1$
Hybrid Mode
$\mathbf{Frequency} \mathbf{Range} \mathbf{Hz}$		$\mathit{\beta } \mathbf{N}$	$\mathit{c} {\mathbf{s}}^{-1}$	${\mathit{R}}_{\mathit{l}} \mathbf{\Omega }$
I	$f_{x1}\le 1.9$	$0$	$0$	$11$
II	$1.9<f_{x1}\le 2.7$	$20$	$10$	$25$
III	$2.7<f_{x1}\le 5.7$	$0$	$0$	$11$
IV	$5.7<f_{x1}\le 9.0$	$0$	$0$	$1$
V	$9.0<f_{x1}$	$0$	$0$	$2$

. The VRS was investigated in a test rig (see Figure 1) by a sinusoidally varying $z$ with amplitude $A_z=1.5 \mathrm{mm}$ and slowly increasing frequency $f$ from $0.1$to $10 \mathrm{Hz}$. Figure 11a,b shows the test results for both modes. The plot in grey refers to semi-active mode and in red to hybrid mode. In addition, the results in semi-active mode with a fixed value of $R_l=1 \mathsf{\Omega }$, marked in green, and $R_l=25 \mathsf{\Omega }$, marked in blue, are compared.

The simulation data in semi-active mode show that the analyzed frequency $f_{x1}$ maybe divided into ranges I, II and III (see Table 4). In the range I and III, it is preferable to assume the smallest value of $R_l$. In range II, the value of the $R_l$ should be as large as possible. The test results confirm that VRS is able to operate without an external energy source.

In hybrid mode, the frequency range may be divided into ranges I, II, III, IV, and V (see Table 4). The values of coefficients $\beta$, $c$, and $R_l$ assumed in the range I, III, and V directly correspond to the parameters as in Section 4. In range IV, due to hardware limitations, the value of $R_l$was assumed to be $1 \mathsf{\Omega }$. In range II, the coefficients $\beta =20 \mathrm{N}$, $c=10 {\mathrm{s}}^{-1}$, and $R_l=25 \mathsf{\Omega }$ were introduced. The high value of the resistance $R_l$ minimizes the energy loss in the potentiometer. The value of parameter $c$ was assumed according to numerical simulations. The value of the parameter $\beta$ was chosen to present that VRS is able to operate with powering from the limited energy source. According to this, the initial increase in the displacement transmissibility $T_{x1z}$ value in the range II results from the small value of parameter $\beta$. Assuming a high value of the parameter $\beta$ lead us to unnecessary energy losses in the linear motor and also requires additional filters to prevent against chattering effect.

6. Conclusions

In the study, dynamic performance and power consumption in 2-DOF VRS with a linear motor were investigated. Simulation studies of the VRS in passive, semi-active and active mode were carried out. In passive and semi-active mode, the effect of additional resistance $R_l$ on the equivalent damping coefficient $c_m$ was considered. In active mode, the control signal $u_m$ which determines the amount of energy supplied from an external source was determined. Based on the simulation data, VRS in hybrid mode operation was proposed.

The analysis of the simulation results led us to the following conclusions:

The effectiveness of VRS in passive and semi-active mode depends on the equivalent damping coefficient $c_m$. Therefore, at low frequencies $f$ it is advantageous to adopt a low resistance $R_l$ (see Figure 4).

In passive and semi-active mode, taking a small $R_l$ results in a reduction in recoverable energy $E_{dy}$. The amount of this energy can be optimised by matching the resistance $R_l$ to the frequency $f$ (see Figure 8).

The effectiveness of VRS in active mode is highest at low frequencies $f$ up to $2.8 \mathrm{Hz}$.

In active mode, the amount of energy consumed significantly exceeds the amount of energy that can be recovered in other modes.

Analysis of $D_p$, $E_c$ coefficients and $T_{x2z}\left(f\right)$ indicates the validity of using the semi-active mode to change the resistance $R_l$ for frequency $f$ up to $2.3 \mathrm{Hz}$.

In the assumed frequency range $f$, five ranges were separated. The principle of hybrid mode operation was proposed by checking the effectiveness of the VRS in each of these ranges.

The slide mode control algorithm implemented in frequency range II of hybrid mode can be improved by using anti-chattering filters or setting the $\beta$ value as a function of frequency.

Further work will aim to develop a new VRS powered by the energy harvested from vibrations. For this purpose, an energy conditioning sub-system will be developed in conjunction with a buffer energy storage and low-energy control system.