With reference to the setup and LS-DEA prototypes presented in Section 4
, this section presents an overview and discussion of the experimental results for the different classes of tests including isopotential, isometric, and isotonic conditions. Different LS-DEA prototypes (Table 2
) were characterized providing evidence of the SR LS-DEA capability to accomplish actuation tasks and validating the design procedure illustrated in Section 3
5.1. Isopotential Tests
Results of isopotential tests provide a measure of the LS-DEA steady-state response, allowing identifying suitable operating ranges where the actuator exhibits a low stiffness (flat force-displacement response) and a stable behavior and to make forecasts on the potential actuation performance.
Isopotential tests were performed on all of the LS-DEA geometries presented in Table 2
. Different tests were performed using a constant frequency
Hz for the prescribed displacements. At such a low frequency, the actuator behavior can be considered quasi-static, the inertial forces being negligible with respect to elastic and electrostatic forces. The displacement amplitude
(crest-to-trough) was varied throughout the different tests within the following set of values: 50, 70, and 90 mm, using a value of
mm for the minimum LS-DEA aperture in each test. For each amplitude value, different tests were performed with different values of the prescribed electric potential, in the range 0–7 kV. The maximum considered voltage was chosen to guarantee a wide safety margin with respect to the electrical breakdown condition (see Figure 2
With reference to tests on prototype LS-DEA #1 and to an amplitude of
mm, Figure 5
A shows the force-displacement responses registered at different voltages.
The following can be observed:
Consistent with the design assumptions, the force-displacement responses show a nearly-flat trend (i.e., a low value of the actuator stiffness) over a significant portion of the considered working range.
Applying voltages within the considered range produces significant variations in the force-displacement response of the actuator. Based on these measurements, electrically-induced force variations of up to 2 N might potentially be obtained, assuming to lock the actuator at intermediate positions within the working range.
The LS-DEA specimen has an inelastic response, due to the SR viscosity and hysteresis, as observed by [3
]. A quantitative analysis of the associated dissipation is presented in the following.
With reference to prototype LS-DEA #3, Figure 5
B compares the prototype force-displacement response at different values of the maximum deformation for purely mechanical tests (
) and at the maximum applied voltage (
kV). Although the prototype cyclic response naturally depends on the maximum deformation amplitude, the sample showed a repeatable force-deformation trend, independent of the displacement amplitude, during the extension phase. In the presence of large voltages, the electrostatic stresses caused the material to lose mechanical tension and the actuator force to fall to zero at small values of coordinate x
, close to
A comparison of the isopotential response of the different LS-DEA prototypes (featuring different pre-stretches) is shown in Figure 6
, for the two cases with applied voltage equal to 0 (Figure 6
A) and 7 kV (Figure 6
B). In the considered range of values, increasing the transversal pre-stretch
leads to a progressive reduction in the LS-DEA mechanical stiffness and, hence, a reduction in the LS-DEA restoring force. Interestingly, the different branches (loading-unloading) of the force-displacement responses have a monotonic increasing trend in the prototype LS-DEA #1 (featuring the lowest pre-stretch), whereas they show a decreasing trend (negative stiffness) at large values of x
for the other two prototypes (with larger transversal pre-stretch). Such a negative stiffness response represents a potential benefit in applications in which the LS-DEA is coupled to stiff mechanical systems, but it might lead to instability issues in the presence of highly variable or randomly applied loads.
The presented isopotential test results suggest that the LS-DEA made of SR is subject to a relevant amount of hysteresis. An estimate of the hysteresis losses associated with the LS-DEA samples’ cyclic stretching is presented in Table 3
. With reference to isopotential tests with
mm, the hysteresis loss was calculated as the ratio of the energy dissipated in a tensile cycle (i.e., the area enclosed by the stabilized force-displacement curve) and the mechanical work spent to achieve maximum deformation (i.e., the area subtended by the loading curve over the entire deformation range). Hysteresis losses over 20% and up to 45% were recorded in the different isopotential tests. For the same SR material, hysteresis losses in the range 9–12% were reported in [3
] for pure-shear tests conducted over a wide range of cyclic operating stretches and strain rates with no applied voltage. In order to understand the origin of the higher hysteresis losses measured in the present tests, let us ideally compare an LS-DEA sample and a pure-shear DE sample that, in a reference configuration, have the same longitudinal and transversal stretches (thus, the same stresses). Owing to the peculiar LS-DEA kinematics (namely, the concurrent contribution of the longitudinal and transversal stresses on the total force), the resulting longitudinal force (per unit cross-section surface of the DE membrane) of the LS-DEA in the actuation direction is lower than that of the pure-shear sample. As a result, the mechanical work (per unit elastomer volume) required to accomplish a given variation in the longitudinal stretch (i.e., the input mechanical energy density) is larger in the pure-shear case. The dissipated mechanical energy density due to hysteresis/viscosity is, in contrast, similar for the two kinematics, or slightly larger in the LS-DEA case (where both the stress variations in the x
directions contribute to mechanical losses). This naturally results in larger values of the hysteresis loss in the LS-DEA, compared to a pure-shear system. In other words, the effect of mechanical stiffness reduction introduced by the transversal stresses in the LS-DEA has the drawback of leading to an increase in the weight of the mechanical losses on the actuator energy balance. Table 3
confirms that increasing the transversal pre-stretch leads to an increase in the hysteresis loss, due to the resulting decrease in the total mechanical stiffness. Similarly, increasing the voltage in isopotential tensile tests leads to an increase in hysteresis loss, as a result of a decrease in the actuator stiffness due to the Maxwell stress.
In actuation tasks, large hysteresis might lead to a significant dissipation of the supplied electrical energy input, limiting the producible useful mechanical energy output. Moreover, the hysteretic behavior might introduce parasitic effects (such as stress relaxation and equilibrium position drift), which are scarcely predictable and may complicate the actuator controllability. A possible solution to overcome potential controllability issues in SR LS-DEAs is the development of advanced control strategies, based on comprehensive numerical models including the effect of electro-mechanical losses [32
] or on machine learning techniques [33
Finally, it is worth noticing that the experimental force-displacement curves measured in isopotential tests (Figure 5
and Figure 6
) fall within the ranges identified in Figure 2
, both in terms of trends and numerical values, despite the uncertainties in the actual values of the prototypes features (e.g., pre-stretches) due to the employed manual manufacturing procedures. The presented mathematical model can thus be employed as an effective design tool to identify suitable combinations of the design ranges aimed at achieving a desired response and can be used as an initial framework to develop advanced controls.
5.2. Isometric Tests
Isometric tests were conducted by commanding time-varying electrical activation of the DEA while keeping the position of the lozenge-shaped mechanisms fixed in prescribed configurations. In these conditions, it was possible to observe the electromechanical response of the DEA without the contribution of the inertia of the four-bar mechanism.
Different isometric tests were performed, using a sinusoidal voltage excitation with an amplitude of 6 kV and frequency f
in the range 1–10 Hz. It is worth noticing that, since the electrostatic component of the LS-DEA force (
in Equation (5
)) depends on the squared voltage, the resulting LS-DEA force variations had twice the frequency of the prescribed voltage variations (i.e., in the range of 2–20 Hz). The upper bound of the frequency range used in the tests was related to setup limitations (namely, the maximum current that the connections between the compliant electrodes and the circuit wiring could safely withstand). It is worth remarking that such a frequency covers a significant portion of the typical/expected working range for DEAs operating at large actuation strains [3
With reference to prototype LS-DEA #3 and to blocking position
mm, Figure 7
A shows the time-series of the force variation generated by an electrical excitation with
Hz. As expected, the LS-DEA force was minimum when the voltage was maximum (positive) or minimum (negative), while it was maximum when the voltage was equal to zero. No phase shift between the force troughs and the voltage maxima/minima was practically present, and a minimum force approximately equal to the initial LS-DEA force (before electrical activation) was reached at each cycle. This proves that, in the considered working frequency range, electrical dynamics can be considered approximately instantaneous. This observation is confirmed by Figure 7
B, which, with reference to the same LS-DEA sample, shows that the cyclic crest-to-trough force variation was practically independent of the force fluctuation frequency. This latter result also suggests that the DE material properties (e.g., the dielectric constant) do not significantly vary within the considered frequency range, thus allowing the achievement of constant blocking forces over a wide frequency range.
The results obtained in the isometric tests can be compared with those from isopotential tests. Figure 8
shows the isopotential
kV and the experimental measurements (at the same voltages) obtained in the isometric tests, for prototypes LS-DEA #2 (Figure 8
A) and LS-DEA #3 (Figure 8
B), respectively. In the plots, the isometric forces in the electrically active state were calculated as the average over the different testing frequencies, due to the low variability of the electrostatic force with the frequency.
The data show that the force measured in isometric tests falls between the loading and the unloading curves of the corresponding tensile cycle, i.e., it is lower than the force measured during the extension phase in the isopotential tests. This is due to the time allowed in isometric tests after the achievement of the blocking position (prior to the voltage application), which causes a relaxation in the LS-DEA membrane force. The electrically-induced force variation in isometric tests is consistent with the shift between the isopotential curves at and kV, hence confirming the capability of the isopotential maps to predict the LS-DEA performance in blocking conditions reasonably.
5.3. Isotonic Tests
Isotonic tests are an effective tool to evaluate a transducer’s performance against a standard task (e.g., lifting a load, making work against a constant force) and to evaluate its dynamics and working bandwidth. For the specific case of the SR LS-DEA, given the significant hysteresis losses observed in isopotential tests, isotonic tests are especially crucial to understand the actual limits in terms of the convertible energy.
For isotonic tests, attention was restricted to LS-DEA #1, i.e., the design with the lowest value of the transversal pre-stretch. Based on the isopotential test results shown in Figure 6
, this implementation was expected to feature a stable response (i.e., positive slope of the
static curves) at constant applied forces over the whole operating range, differently from designs LS-DEA #2 and LS-DEA #3 (which might behave unstably at large loads). Experiments were performed considering three different scenarios: (1) LS-DEA subject to the action of its own weight, with no additional hung mass; (2) LS-DEA with a hung mass
g; and (3) LS-DEA with a hung mass
g. The resulting applied forces in the three cases were (1)
N (equal to 1/2 of the total actuator weight), (2)
N, and (3)
N. In each test, the voltage was cyclically varied between 0 and 5 kV following a sinusoidal profile. Different tests were performed varying the frequency in the range 1–4 Hz.
With reference to two tests with different values of the electric excitation frequency, Figure 9
shows the time series of the LS-DEA displacement induced by the electric excitation (the initial equilibrium position x
and the slow position drifts due to rubber stress relaxation were subtracted from the time series to facilitate the comparison of different scenarios). The following could be observed:
At the lowest frequency (Figure 9
A), the actuator operation can be considered quasi-static. As a consequence, the LS-DEA displacement was in phase with the electrical excitation. At the time instants where the voltage equaled zero, the device reached a position that approximately equaled the initial equilibrium position (i.e.,
). The oscillation amplitude increased with the applied load (hung mass), owing to the lower stiffness of the LS-DEA at large applied forces (see the slope of the
curves in Figure 6
Increasing the excitation frequency (Figure 9
B) leads the LS-DEA to operate closer to its mechanical resonance frequency. As a consequence, the oscillation amplitudes in the scenarios with larger hung mass increase (compared to Figure 9
A) and negative values of
The increase in the oscillation amplitude was maximum in the case 180 g, while it was practically negligible for . This is easily explained in terms of the actuator natural frequency in the different scenarios. The higher the applied weight on the device, the lower the natural frequency is, due to two combined effects: (1) the increase in the actuator inertia; (2) the decrease in the LS-DEA stiffness. In the case , the actuator still behaved in a quasi-static way, whereas in the case g, it showed a nearly-resonant behavior, as demonstrated by the significant phase shift between the excitation signal and the displacement.
The displacement time series significantly divert from the sinusoidal trend, owing to the highly nonlinear LS-DEA response and to the presence of low-amplitude parasitic oscillation modes (e.g., rotation above the lozenge mechanism fixed hinge), which led to the presence of multiple peaks within the same oscillation.
An overview of the LS-DEA dynamical response over the considered frequency range is shown in Figure 10
. The figure reports the total oscillation amplitude measured at the different frequencies.
In the scenarios with larger applied mass, the resonance frequency (at which the oscillation amplitude is maximum) falls within the investigated frequency range, namely around Hz for g and Hz for g. In the case with no hung mass, the natural frequency lies above the upper bound of the considered range, as shown by the slight increase in the oscillation amplitude with frequency. The maximum amplitude was reached in the presence of the largest mass, as in this case, the equilibrium position of the system falls in a region where the working range is characterized by lower mechanical stiffness. In this context, a maximum stroke of 22 mm was reached, corresponding to 18% the lozenge mechanism side length.
Remarkably, despite the large hysteresis observed in isopotential tests, the LS-DEA was capable of effectively accomplishing the isotonic actuation task, providing an energy density (gravitational energy variation of the hung mass in a semi-oscillation per unit DE mass) of up to 20 mJ/g in quasi-static conditions (
Hz) and of 50 mJ/g around the resonance condition. The previous figures significantly increase to 50 mJ/g at 1 Hz and 140 mJ/g at resonance if the gravitational energy variation associated with the lozenge mechanism are also considered, leading to results comparable with those demonstrated with soft acrylic DEs [34
]. Remarkably, this result was achieved with a DEA embodiment that did not make use of any external stiffness cancellation mechanical element, as opposed to most DEA layouts based on high-modulus elastomers (e.g., silicones) proposed so far [14
Large margins of improvement for the SR LS-DEA are possible, both in terms of convertible energy density and operating bandwidth, if an optimized lightweight design for the system (e.g., based on a fully polymeric compliant frame) is pursued.