Design and experimental characterization of a bypass magnetorheological damper featuring variable stiffness and damping

Figure 1 shows the schematic of the proposed VSVD-MRFD with an annular-radial bypass valve. The VSVD-MRFD composes of main piston, piston rod inside a cylindrical housing, two springs with different stiffness and an external bypass MRV. The cylindrical housing is divided by the main piston into upper and lower chambers that are filled with MR fluid. The housing is connected with an end cap and cylinder cover at both of its ends. The main piston is threaded with the piston rod and is sealed with the cylindrical housing. The piston rod is connected to the external mechanical spring, and the end cap is attached to the cylindrical housing by a sliding guider. The piston rod has an internal hollow chamber with certain dimensions which accommodates a floating piston connected to a mechanical spring. This chamber with the internal mechanical spring basically serves as an accumulator to compensate the change in the MR fluid volume due to the movement of the piston rod. Compared with the conventional bulky gas accumulator (gas spring), the proposed design yielded a compact and reliable MRFD with no issues related to gas leakage and fluid–gas dynamics. The external bypass MRV, shown by the dashed line in figure 1 consists of a housing, MRV bobbin with the embedded magnetic coil and a spacer to allow annular-radial fluid flow. The annular-radial bypass MRV provides a simple design which causes the MRFD to generate high dynamic force and dynamic range and facilitates also the installation and maintenance. Moreover, the bypass MRV also provides better heat dissipation compared with internal MRVs under high applied currents, thus allowing the damper to operate for a longer time under harsh environmental conditions.

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The working principle of the proposed VSVD-MRFD is based on the series connection between the damping of the MRV unit and the spring element as demonstrated in figure 2. The series connection type provides an easy and practical implementation of the VSVD-MRFD in real world applications. The damping variability is realized by controlling the applied magnetic field to the annular-radial bypass MRV. Since the damping element is in series with the internal spring k₁, the equivalent damping is not depending only on the damping of the MRV but also, it depends on the stiffness of the internal spring and excitation frequency, as represented in equation (1). The stiffness variability of the proposed damper is also achieved via series connection of MRV unit and the internal spring (k₁) which are in in parallel with the external spring element k₂. By adjusting the applied current, the force applied on the internal and external springs will be varied. The equivalent stiffness (${k_{{\text{eq}}}}$), and equivalent damping (${C_{{\text{eq}}}}$) of the designed MRFD are given in equations (1) and (2), respectively.

$$\begin{equation}{C_{{\text{eq}}}} = \frac{{ck_1^2}}{{{c^2}{\omega ^2} + k_1^2}}\end{equation} \tag{ 1 }$$

$$\begin{equation}{k_{{\text{eq}}}} = {k_2} + \frac{{c{k_1}{\omega ^2}}}{{{c^2}{\omega ^2} + k_1^2}}.\end{equation} \tag{ 2 }$$

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The basic geometry of the fabricated VSVD-MRFD is shown in figure 3 and its main dimensions are listed in table 2. Figures 4(a) and (b) show the components and the prototype of the VSVD-MRFD, respectively. The housing cylinder of the damper is fabricated from heat-treated alloy steel 4140 while the carbon steel 1045 was used in the manufacturing of the main piston, piston rod and floating piston. A bronze guider was used to connect the piston rod with the housing cylinder, thereby providing sufficient lubricant for the piston rod during movement. For reducing the magnetic circuit reluctance, a low carbon steel AISI 1117 was used as the material of the bobbin and steel AISI 1006 was used for the shell of the bypass MRV. The spacer material between the bobbin is Nylon 6/6 in the annular gap and stainless-steel SS 304 in the radial gap. These materials properly guide the magnetic flux lines toward active areas in the radial and annular fluid flow channels. Lord Corporation's MRF-132DG MR fluid was utilized to fill the MRFD. The specific gravity of the MR fluid is 3, and its viscosity is 0.112 Pa.s at 40 °C. The electromagnet of the MRV has 215 turns of the copper wire of American Wire Gauge 24. Two heavy duty springs from Lee Company with different stiffness coefficients were utilized to provide stiffness variability. Table 3 shows the detail parameters of the two springs.

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Table 2. Main parameters of the proposed VSVD MRFD.

Parameter	Symbol	Value (mm)
Duct gab	$d$	1.2
Coil width	${w_{\text{c}}}$	7.0
Piston diameter	${D_{\text{p}}}$	70.0
Piston rod diameter	${D_{\text{r}}}$	30.0
Floating piston diameter	${D_{\text{f}}}$	19.05
Radial duct outer diameter	${D_{\text{o}}}$	38.1
Radial duct inner diameter	${D_{\text{i}}}$	10
MR valve whole diameter	$D$	64.5
Annular duct length	${L_{\text{a}}}$	43.9
Coil length	${L_{\text{c}}}$	10.7
MR valve height	$L$	60.0

A comprehensive experimental investigation was conducted to evaluate the dynamic performance of the proposed bypass VSVD-MRFD with an annular-radial gap in terms of force-displacement and force-velocity characteristics under a relatively wide ranges of mechanical and magnetic loading conditions. The designed test rig and the specifications utilized in the measurements have been thoroughly explained in the previous study [63]. Briefly, figure 5 shows the full test setup for implementing the dynamic characterization of the VSVD-MRFD. The piston of the damper is attached to the load cell from the top and the lower part of the damper is connected to the lower part of an electro-hydraulic actuator. The input displacement is provided by the actuator and the resulting force is recorded from the load cell. The measured data were then transferred to the PC shown in figure 5, via a data acquisition system (National Instrument Inc). A power supply was employed to provide the required electrical current for the MRV.

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The experimental measurements were conducted to measure force-time, force-displacement, and force–velocity of the prototyped MRFD under various input motion amplitudes and frequencies, and excitation currents. In particular, the experiments were performed with five levels of excitation frequency (f = 0.5, 1, 2, 3, and 4 Hz), two levels of displacement amplitude (X = 1 and 2.5 mm), and six levels of applied current (I = 0, 0.25, 0.5, 1, 1.5 and 2 A). For each test condition, force-displacement and force-velocity were measured in five consecutive cycles after observing the steady-state data from the LABVIEW software and then averaged for characterization.

The nonlinear dynamic hysteresis characteristics of the prototyped VSVD-MRFD under a wide range of loading conditions are provided and discussed in this section. Figure 6 shows the force-displacement and force-velocity characteristics of the VSVD-MRFD at the excitation frequency of 0.5 Hz, and displacement amplitude of 1 mm. The results are presented for different applied electrical currents, ranging from 0 to 2 A. Results show that the force-displacement hysteresis characteristics and the force-velocity curves become more noticeable and enhanced with increasing the applied current. As it can be realized, the area enclosed by the force-displacement hysteresis loops, which represents the amount of energy dissipation per cyclic loading, is substantially increasing with increasing the applied current. This is also confirmed by the slope of the main axis of the force-velocity curves, which correspondingly denotes the equivalent damping, which is increased by increasing the applied current. Similar behavior was also observed at other loading conditions. Figure 6 further demonstrates that under this loading condition, the peak force is increased from about 1.53 kN in the absence of the current to nearly 5.44 kN under current of 2 A, thus yielding a dynamic range of nearly 3.6. The magnetic saturation phenomenon can also be observed as the applied current exceeds 1.5 A. Furthermore, figure 6(a) also shows that the equivalent stiffness of the MRFD increases as the applied current increases from zero to 2 A. The equivalent stiffness represents the slope of the lines that connect the bottom-left point of the hysteresis curves to the top-right point, corresponding to the force at minimum displacement and the force at maximum displacement, respectively.

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Figure 7 exhibits variation of the dynamic characteristics of the VSVD-MRFD with respect to applied current at the excitation frequency of 1 Hz and displacement amplitude of 2.5 mm. At this loading condition, the damper's peak force ranges from 2.0 kN to 7.2 kN as the applied current ranges from zero to 2 A, yielding also a dynamic range of 3.6. Results consistently confirm that the enclosed area by the force-displacement loops (energy dissipation) amplifies as the applied current increases. Figures 6(b) and 7(b) also demonstrate that the damping force increases with the increasing velocity of the damper piston rod.

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Figure 8 illustrates the effect of applied current on the dynamic characteristics of the VSVD-MRFD. Results are presented for the excitation frequency of 2 Hz and displacement amplitude of 1 mm. At this loading condition, the damper's peak force ranges between 1.64 kN and 6.73 kN as the applied current ranges from zero to 2 A, representing a higher dynamic range of 4.1. Figures 7(a) and 8(a) also reveal that the slope of the major axis in the force-displacement loops, which corresponds to the equivalent stiffness of the damper, increases as the applied current increases from zero to 2 A, thereby confirming the field-dependency of the stiffness. Moreover, it should be noted that the loading (ABC) and unloading (CDA) paths possess different instantaneous slopes. It is because that the proposed damper has two spring elements, in which when the piston moves within the loading path (e.g. from minimum displacement to zero intermediate position), the spring elements are compressed, thereby varying the slope of loading curves in a piecewise manner (i.e. see the path of AB in figure 7(a)). Increasing the applied current increased the aforementioned slope, as shown in figure 7(a). However, such variation with respect to current was not observed during the unloading path (CD & DA), as expected. The force-displacement curves presented in figure 7(a) show the same slope trends for the unloading path (CD&DA), irrespective of the applied current. Nonetheless, the zero-current force-displacement path shows different path during loading curve (i.e. AB). It is because that the stiffness variation is highly dependent on the applied current. Relatively, similar trends were also observed for other loading conditions, as shown in figure 8(a)

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Figures 9 and 10 reveal the effect of excitation frequency on the dynamic properties of the VSVD-MRFD in terms of force-displacement and force-velocity characteristics at a constant displacement amplitude of 1 mm under an applied current of 1 A and 2 A, respectively. Results show as the loading frequency increases, the slope of the force-velocity curves, representing the equivalent damping coefficient of the MRFD, decreases. This is due to the fact the equivalent viscous damping is inversely related to the excitation frequency as it will be addressed and quantified in section 4.3. The peak value of the damping force which occurs around zero displacement or near maximum velocity, initially increases noticeably by increasing the loading frequency from 0.5 Hz to 2 Hz, as can be seen both in the force-displacement and force-velocity characteristics (although at different peak velocities) in figures 9 and 10. The peak damping force, however, slightly increases by increasing the loading frequency from 2 Hz to 4 Hz. This is expected as the percentage of increase in the frequency from 0.5 to 2 Hz (300% increase) is substantially more than that from 2 to 4 Hz (100% increase) and also by increasing the frequency the equivalent viscous damping coefficient decreases. It should be also noted that increase in the peak damping force due to increase in the frequency is very small compared with that due to increase in the applied current. This is due to the fact that the excitation frequency affects the viscous damping force which is relatively very small compared with the controllable yield force due to the applied current. As example, the maximum damping forces under displacement of 1 mm, applied current of 2 A and loading frequency of 3 Hz is 6.8 kN which increases to 7.2 kN, by increasing the frequency to 4 Hz.

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Figure 10 presents the frequency dependent characteristics of the VSVD-MRFD when 2 A is applied to the MRV's coil. Results show that the excitation frequency yields more nonlinear hysteresis effect on both force-displacement and force-velocity characteristics of the VSVD-MRFD, when higher levels of currents (i.e., 2 A) are applied in comparison to the lower levels of current (i.e., 1 A), as presented in figure 9. For instance, at a higher frequency of 4 Hz, the loading paths of the hysteresis loops presented in figure 10(b), tend to be upward, enhancing the intra-cycle damping coefficient locally. Further analysis of figures 9 and 10 also reveals that increasing the applied current, and thus the magnetic field within the MRV leads to a substantial increase in the energy dissipation properties and damping force of the VSVD-MRFD. Figures 9(a) and 10(a) further reveal that the slope of the force-displacement curves, representing the equivalent stiffness coefficient of the MRFD, slightly increases. It should be noted that the loading and unloading paths of the force-displacement curves relatively show similar trends as observed for non-zero current force-displacement profiles presented in figures 7(a) and 8(a), irrespective of loading frequency. For instance, when the loading frequency increases from 1 Hz to 4 Hz, the slope of the loading path increases, due to the added effect of frequency-dependent friction force of sealing materials within the cylinder-piston assembly, as shown in figures 9 and 10. This can enhance the stiffness of the proposed damper, as the applied current increases within this range.

Figures 11 and 12 explain the displacement dependent properties of the fabricated VSVD-MRFD in terms of force-displacement and force-velocity under excitation currents of 0.5 A and 1.5 A, respectively. Here, these characteristics were studied at a constant excitation frequency of 1 Hz as an example. Results show that the energy dissipation properties and damping force of the VSVD-MRFD is considerably increased by increasing the displacement amplitude from 1 mm to 2.5 mm. Figures 11(a) and 12(a) clearly show the increase in the peak force as the loading displacement increases from 1 mm to 2.5 mm. Figures 11(b) and 12(b) also manifest that the loading paths of the force-velocity curves are relatively independent of the excitation displacement. Similar to figures 9 and 10, the slope of the main axis of the force-velocity loops, which represents the equivalent damping coefficient of the MRFD, tends to decrease with increasing displacement amplitude. The slope of the major axis of the force-displacement loops, which represents the equivalent stiffness, also decreases as the displacement amplitude increases from 1 mm to 2.5 mm.

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Figure 13 shows the force-displacement and force-velocity characteristics of the prototyped MRFD under a loading frequency of 4 Hz and displacement amplitude of 2.5 mm. It should be noted under these loading conditions, the maximum force generated by the bypass MRFD was reached to as high as 7.82 kN. The damper could not be tested at higher levels of motion frequencies and amplitudes due to the limitation of the electro-hydraulic actuator system and the load cell capacity. Examination of results also reveal that the force-velocity curve presented in figure 13(b) shows secondary or sub-hysteresis loops (cross-over points) at the end of the loading cycle, which is observable at the top-right of the force-velocity curves, irrespective of applied current. These secondary hysteresis loops correspond to peaks in the loading path, which can be seen before and after the intermediate position of zero displacement. It is partly due to the fact that the peak damping force occurred before the instance of the maximum velocity. This implies that the peak force location occurs before the instance of the maximum velocity. This may in part be due to the friction force generated by mechanical friction between the seals and the shaft/piston, as has also been observed in other studies [64–66] and also possibility of slight rocking motion (beside vertical motion) due to the higher frequency vibration. Further, this behavior may also be due to the inertia effect of the MR fluid (1000 ml of MR fluid is contained within the damper) [67, 68], as well as entrapped air in the damper [69–71].

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The presentation of the remaining experimental data, including the force-displacement, force-velocity under loading frequencies of 0.5–4 Hz, applied current of 0–2 A, and displacement amplitudes of 1 and 2.5 mm, are presented in Figure 19, 20, 21, 22, and 23 in the appendix.

The presented results provide a qualitative analysis of the prototyped VSVD-MRFD. The following subsections will present quantitative assessments of the fabricated VSVD-MRFD in terms of dynamic range, equivalent viscous damping coefficient, and equivalent stiffness coefficient. An investigation of these indices under various loading conditions and magnetic fields is also discussed.

The dynamic range is an important figure of merit, indicating the bandwidth and control performance of VSVD-MRFDs to mitigate vibrations under different environmental conditions. The dynamic range is defined as the ratio between the maximum damping force observed at the maximum current applied (I_max) over the minimum damping force at off-state conditions (I = 0 A) [72]. The dynamic range ($\lambda$) can, thus, be calculated as:

$$\begin{equation}\lambda = \frac{{{F_d}_{{I_{{\text{max}}}}}}}{{{F_d}_{I = 0{\text{A}}}}}\end{equation} \tag{ 3 }$$

where ${F_d}_{I = 0{\text{A}}}$ is measured damping force under zero magnetic field, while ${F_d}_{{I_{{\text{max}}}}}$ is the measured damping force under maximum applied current. The dynamic range was evaluated under the entire range of the mechanical and magnetic loading conditions considered. Figures 14(a) and (b) illustrate the influence of the excitation displacement and loading frequency on the evaluated dynamic range under different applied currents, respectively. Results clearly reveal that for the given loading condition, the dynamic range substantially increases as the magnetic field increases. This is consistent with the increment in the peak force of the force-displacement observed in figures 6–8. Further examination of results shows that the dynamic range decreases at higher levels of the displacement amplitude and loading frequency, while the influence of displacement is more obvious than that of loading frequency. This is in part due to the greater off-state damping forces at higher loading frequencies and displacement amplitudes, which decreases the dynamic range. Results show that the damper can reach to a maximum dynamic range of 4.5 at the loading frequency of 1 Hz and a displacement amplitude of 1 mm.

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As discussed in section 4.1, the VSVD-MRFD characteristics can be quantified either using the enclosed area of the force-displacement curves (the equivalent viscous damping coefficient (${C_{{\text{eq}}}}$)) using the energy dissipation method or the slope of the major axis of the force-velocity. The energy dissipation method has traditionally been employed for evaluating the equivalent viscous damping coefficient (${C_{{\text{eq}}}}$) at a certain frequency ($f$). The energy dissipated over one cycle ($U$) can be calculated from the enclosed area of the force-displacement curve as follows [73]:

$$\begin{equation}U = \mathop \oint \limits_0^{2\pi /\omega } F\left( t \right)dx = \mathop \int \limits_0^{2\pi /\omega } F\left( t \right)v\left( t \right)dt\end{equation} \tag{ 4 }$$

in which $F\left( t \right),\,v\left( t \right)$, and $x$ represent output force, velocity, and displacement, respectively. The equivalent viscous damping coefficient can, thus, be evaluated as [73]:

$$\begin{equation}{C_{{\text{eq}}}} = \frac{U}{{2{\pi ^2}f\,X_0^2}}\end{equation} \tag{ 5 }$$

where ${X_0}$ is the displacement amplitude of the input excitation. The ${C_{{\text{eq}}}}$ can be also estimated from the slope of the major axis of the force-velocity curves by finding the damping forces ${\left[ {{F_d}} \right]_{{v_{{\text{max}}}}}}$ and $\,{\left[ {{F_d}} \right]_{{v_{min}}}}\,$ corresponding to the maximum and minimum piston velocity values ${v_{{\text{max}}}}$ and ${v_{{\text{min}}}}$ respectively as:

$$\begin{equation}{C_{{\text{eq}}}} = \frac{{{{\left[ {{F_d}} \right]}_{{v_{{\text{max}}}}}} - {{\left[ {{F_d}} \right]}_{{v_{{\text{min}}}}}}}}{{{v_{{\text{max}}}} - {v_{{\text{min}}}}}}\end{equation} \tag{ 6 }$$

The equivalent viscous damping was evaluated using the above two approaches for a loading frequency of 2 Hz and was summarized in table 4, as an example. Results show that the equivalent viscous damping evaluated using both approaches are nearly comparable. Such comparison has been rarely reported in the literature. It was found that the maximum relative difference between the two approaches was almost 2.1%. Furthermore, increasing the excitation displacement leads to a decline in the ${C_{{\text{eq}}}}$ confirming the observation in figures 11 and 12. This decrement is also consistent with the force-displacement and force-velocity characteristics presented in figures 6–8.

The influence of loading displacement and frequency on the variation of the ${C_{{\text{eq}}}}$ versus applied current is shown in figures 15 and 16. For instance, figures 15(a) and (b) show the variation of the calculated ${C_{{\text{eq}}}}$ based on the two approaches, considering different displacement amplitudes of 1 mm and 2.5 mm, as a function of applied current. Results indicate a significant decrease in the ${C_{{\text{eq}}}}$ as displacement amplitude increases. This is consistent with the qualitative analysis presented in figures 11 and 12.

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Furthermore, figures 16(a) and (b) show the variations of calculated ${C_{{\text{eq}}}}$ versus applied current for various loading frequencies, ranging from 0.5 Hz to 4 Hz, calculated by two aforementioned methods. Results reveal a nonlinear increment in the ${C_{{\text{eq}}}}$ with increasing applied current while it diminishes for higher loading frequency, which confirms the observation in section 4.1 (see figures 9 and 10). For instance, the ${C_{{\text{eq}}}}$ at zero current and displacement amplitude of 2.5 mm under loading frequency of 0.5 Hz and 3 Hz were respectively obtained as 1.69 kN s cm⁻¹ and 0.67 kN s cm⁻¹. The corresponding ${C_{{\text{eq}}}}$ at applied current of 2 A were obtained as 7.18 kN s cm⁻¹, and 1.70 kN s cm⁻¹, respectively. These reveal the relative increments of 425%, and 253% at 0.5 Hz and 3 Hz, respectively, when applied current increased from zero to 2 A. A closer examination of figures 15 and 16 also reveal that the ${C_{{\text{eq}}}}$ is prone to magnetic saturation when applied current exceeds 1.5 A. This is also consistent with saturation magnetization within force-displacement curve (e.g. peak force) observed in figure 7.

The proposed VSVD-MRFD is also characterized to examine its stiffness variability by evaluating the equivalent stiffness of the system. MRFD featuring stiffness variability can be effectively utilized to significantly reduce the transmitted vibration by varying the natural frequency of the system to avoid resonance occurrence. Moreover, it can also overcome the conflict between good handling and ride comfort for off/on-road vehicles [74, 75]. The equivalent stiffness (${K_{{\text{eq}}}}$) of the fabricated VSVD-MRFD can be evaluated from the slope of the axis line intersecting the point at maximum and minimum force-displacement hysteresis cycle (See figure 6). This slope is varied according to the applied current and can be estimated from the following equation as [34]:

$$\begin{equation}{K_{{\text{eq}}}} = \frac{{{{\left[ {{F_{\text{d}}}} \right]}_{{D_{{\text{max}}}}}} - {{\left[ {{F_{\text{d}}}} \right]}_{{D_{{\text{min}}}}}}}}{{{D_{{\text{max}}}} - {D_{{\text{min}}}}}}\end{equation} \tag{ 7 }$$

where ${\left[ {{F_{\text{d}}}} \right]_{{D_{{\text{max}}}}}}$ and ${\left[ {{F_{\text{d}}}} \right]_{{D_{{\text{min}}}}}}$ denote the maximum and minimum forces generated by the damper, respectively, corresponding to the maximum and minimum value of the stroke, namely, ${D_{{\text{max}}}}$ and ${D_{{\text{min}}}}$.

Figures 17 and 18 exhibit the variation of ${K_{{\text{eq}}}}$ of the VSVD-MRFD with respect to the applied current considering different levels of loading displacement and frequency. For example, figures 17(a) and (b) reveal the influence of the excitation displacements of 1 mm and 2.5 mm on the evaluated ${K_{eq}}\,$at loading frequencies of 1 Hz and 2 Hz, respectively as a function of the applied current. The results indicate that the ${K_{{\text{eq}}}}$ significantly increases as the current increases. However, similar to the equivalent viscous damping, the rate of increasing is reduced as the applied electric current increases from 1 A to 2 A indicating the magnetic saturation phenomenon. Moreover, results reveal a remarkable reduction in the ${K_{{\text{eq}}}}$ as the excitation displacement increases. This decrement is also consistent with the observed decrements of the slope of the major axis of the force-displacement in figures 11(a) and 12(a).

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Additionally, figures 18(a) and (b) show the variation of the evaluated ${K_{{\text{eq}}}}$ under various loading frequencies, ranging from 0.5 Hz to 4 Hz, as a function of the applied current, calculated for excitation displacements of 1 mm and 2.5 mm, respectively. Results also reveal a nonlinear increasing in ${K_{{\text{eq}}}}$ with increasing the applied current. Results also show that the increment in ${K_{{\text{eq}}}}$ with respect to current is more pronounced at higher levels of applied frequency. For example, the ${K_{{\text{eq}}}}$ at zero current and displacement amplitude of 1 mm under loading frequency of 0.5 Hz and 3 Hz were respectively obtained as 10.90 kN cm⁻¹, 14.26 kN cm⁻¹. The corresponding ${K_{{\text{eq}}}}$ at applied current of 2 A were obtained as 51.60 kN cm⁻¹, and 69.70 kN cm⁻¹, respectively. These reveal the relative increments of 473%, and 488% at 0.5 Hz and 3 Hz, respectively, when the applied current increased from zero to 2 A. Further observation of figures 17 and 18 reveals that the ${K_{{\text{eq}}}}$ has a tendency to saturate when applied current exceeds 1.5 A.

The ${K_{{\text{eq}}}}$ was quantitatively evaluated for loading frequency of 1 Hz and displacement of 2.5 mm under various applied currents and was summarized in table 5, as an example. Results show the wide range variability of the ${K_{{\text{eq}}}}$ with the applied current. The equivalent stiffness increases from 7.78 kN cm⁻¹ at zero current to almost 27.57 kN cm⁻¹ at the applied current of 2 A, rendering a dynamic range of almost 3.5 in view of stiffness.

Table 5. Equivalent stiffness coefficient under different applied currents.

Applied current	Equivalent Stiffness, ${K_{{\text{eq}}}}$ (kN cm−1)
0	7.78
0.25	11.44
0.5	17.81
1	23.05
1.5	26.47
2	27.57