Biomimetic flexible plate actuators are faster and more efficient with a passive

Peter D.Yeh·Alexander Alexeev

Biomimetic flexible plate actuators are faster and more efficient with a passive attachment

Peter D.Yeh1·Alexander Alexeev1

©The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Using three-dimensional computer simulations, we probe biomimetic free swimming of an internally actuated flexible plate in the regime near the first natural frequency. The plate is driven by an oscillating internal moment approximating the actuation mechanism of a piezoelectric macro fiber composite(MFC)bimorph.We show in our simulations that the addition of a passive attachment increases both swimming velocity and efficiency.Specifically,if the active and passive sections are of similar size,the overall performance is the best.We determine that this optimum is a result of two competing factors.If the passive section is too large,then the actuated portion is unable to generate substantial deflection to create sufficient thrust.On the other hand,a large actuated section leads to a bending pattern that is inefficient at generating thrust especially at higher frequencies.

Flexible plate·Biomimetic·Lattice Boltzmann· Fluid–structure interaction·Robotic fish·Piezoelectric actuator

DOI 10.1007/s10409-016-0592-0

1 Introduction

The development of efficient aquatic machines that mimic the swimming motion of biological fish is an ongoing challenge.Fish leverage the distributed flexibility of their body and fins in order to swim quickly and with great agility [1].Experimental studies have revealed that individual finstrokes are highly complex and mostly three-dimensional [2–5].Conventional aquatic machines employ servomotors or hydraulic actuators in conjunction with a series of linked rigid sections in order to mimic body flexibility in a single bending plane[6–11].However,generating a large range of three-dimensional fin strokes requires a complicated and cumbersome system of links and pulleys.Although well-studied,these motor-based actuators still cannot match the swimming performance of biological fish.

On the other hand,internally actuated(active)smart materials can be adapted to perform complex fish-like motions and controlled to follow different strokes without the physical complexity of motor-based designs.Typical smart materials used for biomimetic swimming devices include ionic polymer–metal composites[12–18],shape memory alloys [19–24],magnetostrictive thin films[25–27],and others [28–30].Among other active smart materials,piezoelectric actuators are attractive because of their geometric scalability,high efficiency,and noiseless performance[31].Despite these advantages,the current understanding and usage of piezoelectric actuators in swimming devices is still limited [32–35].

Macro fiber composites(MFCs)[36–38]are a new class of piezoelectric actuators that have been recently used and tested in small-scale aquatic propulsors[39–41].MFCs offer large dynamic stresses in bending and high performance at low and high frequencies.The thrust performance of MFCs as piezoelectric biomimetic actuators was recently investigated by Erturk and Delporte[41].In this study,the authors fabricated MFC bimorph actuators,which are two MFC laminates bonded together by a thin layer of epoxy.When oppositely signed electric fields are applied to the different layers,one stretches and the other contracts,creating bending.Thus, these flexible MFC bimorph actuators are driven by oscil-lating internal moments.The thrust generation of the MFC bimorph cantilevers was subsequently measured experimentally.Furthermore,these MFC bimorphs were also recently used to build an untethered swimming device that resembled a robotic fish[39].The MFC bimorph acted as the fin,providing oscillating flexible propulsion.A waterproof“body”housed the electronics that powered the bimorphs,allowing for unrestricted movement within a fluid.

Fig.1 Schematic of internally actuated flexible plate with passive attachment.a Geometry of plate showing the dimensions and size of the active portion(darker shade)and passive portion(lighter shade).b Oscillating internal moment in the active section leads to asymmetric bending patterns that generate propulsion

In addition,Erturk and Delporte[41]probed the effect of a passive caudal fin attachment by comparing thrust with and without the passive fin.It was found that the passive fin attachment introduced an additional bending mode in the same frequency range,leading to wideband thrust generation.Furthermore,the thrust amplitude was higher for the cantilever beam with a passive attachment.These results suggest that active,internally actuated fins with and without passive attachments behave differently and that a particular combination of active and passive sections may lead to optimal swimming performance.Thus,the purpose of our study is to use computational simulations to systematically study the performance of biomimetic swimmers with active(via internal moment)actuation with a passive attachment and to identify the physical mechanisms that enhance the swimming performance of the composite active–passive fins.

Simulations facilitate the fundamental understanding of the physics behind biomimetic oscillating propulsion.Computational studies of flexible fins have initially started with simple two-dimensional models with torsional springs[42] or elastic filaments[43–47].Recent advances in computing power have allowed for more complex three-dimensional simulations.Current models are capable of simulating the three-dimensional interaction between the oscillating plate and the surrounding viscous fluid[48–51].Recent computational[52–55]models and also experiments[56–64]have focused on the thrust production and free swimming of flexible plates in order to explore fast and efficient swimming regimes.However,these modeling efforts have exclusively focused on passive fins,whose behavior is qualitatively different from that of internally actuated plates.To the best of our knowledge,the computational study of free swimming of oscillating plates with internal actuation has not yet been undertaken.

Here,we use computational simulations to study the swimming performance of internally actuated flexible plates with passive attachments as a suitable fin actuator for aquatic swimming devices.Experimentally,the active fins,such as those made of MFCs,could be attached to a power housing enclosing electronic components necessary to control fin beating and to enable unrestrained locomotion of the device. To understand the role of the active fin in the swimmer propulsion,we focus on the hydrodynamics of an internally actuated plate(or fin)with a clamped boundary condition imposed at the plate leading edge to mimic its attachment to a power housing.We do not include the housing in our computational model to isolate the effects governing the plate propulsion. We expect that the power housing would impose a drag force resisting the plate propulsion and decreasing the swimming speed.The magnitude of this resistive drag will strongly depend on the particular geometry of the power housing and, therefore,can greatly vary between different experimental realizations.We expect,however,that the basic trends found in this study will hold,thereby providing practical guidelines for designing efficient biomimetic devices that employ active flexible propulsors.

Our model actuator is shown in Fig.1a.The active fin is modeled as a flexible plate of uniform bending stiffness with total length L,width w,and thickness b,which is small compared to the length(b≪ L).The aspect ratio L/w is 2.5.The plate is segmented into two sections.First,an internally actuated section,called the active portion(shown in a darker shade),extends a distance d from the leading edge.Beyond that,a passively responding tail section(the passive portion,lighter shade)is attached.Both sections have the same isotropic material properties.The active portion actuates the swimmer using a sinusoidally oscillating internal moment,M(t)=M0sin(ωt),uniformly distributed within the active portion.Here,ω is the driving frequency, and M0is the amplitude of the applied moment.This actu-ation pattern mimics that of an oscillating bimorph MFC composite[39].

The leading edge vertical displacement and slope are both kept at zero(clamped in the z-direction),but the plate is allowed to swim forward horizontally.The clamped boundary condition mimics an oscillating fin attached to a power housing with a large enough mass to prevent leading edge deflection.The goal of our study is to investigate the effects of adding the passive elastic attachment on the steady state swimming velocity.To isolate these effects,we hold constant the total length of the swimmer and vary the size of the active portion.The plate is driven at constant frequency,but its stiffness is varied.Depending on the proximity to resonance,different bending patterns emerge that lead to faster or slower swimming velocity.

2 Computational model

Our computational model is a fully coupled solver,capturing both the elastic deformation response and the hydrodynamics.The fluid mechanics is modeled using a three dimensional lattice Boltzmann model(LBM),while the solid mechanics is captured by a lattice spring model(LSM).These two models are fully coupled by boundary conditions.This fluid–structure interaction model has been used previously to study the free swimming of elastic plates undergoing plunging actuation[52,53].

The LBM is a particle based mesoscale method that simulates an incompressible Newtonian fluid.The computational fluid domain is discretized into a cubic lattice of equally spaced“fluid nodes”.At each node,continuous velocity distribution functions,fi(r,ci,t),characterize the flow.The distribution functions represent the mass density of“fluid particles”at position r that propagate with velocity c in specifically defined direction i.The distribution function evolves in time according to the discrete Boltzmann equation [65].We employ a D3Q19 model,meaning that a three-dimensional cubic lattice with 19 distribution functions is used.Hydrodynamic fields such as density ρ,momentum j=ρu,and stresses Π are computed as moments of fias follows To capture the solid mechanics of the plate,we employ an LSM[66,67].In this simple model,the continuous elastic thin plate is discretized into a network of masses connected together by harmonic springs arranged on a regular lattice. As shown in Fig.2,we use a triangular lattice,leading to isotropic behavior and Poisson ratio ν=1/3.The stretching springs that connect nodes have stiffness ks.Note that ksis relatively large so the plate is nearly inextensible.In order to model bending,every collinear triplet of nodes defines one bending unit.At the center node of each bending unit is a torsional spring with stiffness kb,which resists out-of-plane bending.On a triangular lattice,this bending spring configuration is related to the one-dimensional beam bending rigidity EI as the following[67,68]

The different types of nodes are also depicted in Fig.2a. On the leading edge(left),the open squares indicate nodes that are prescribed to be fixed in the vertical direction.Having two fixed rows leads to a zero slope boundary condition. The darker circles indicate nodes within the active section, while the lighter circles represent nodes forming the passive section.The division between active and passive sections isindicated by the dotted line.All the nodes to the left are within the active section,while those to the right are within the passive section.Both the active and passive section nodes have the same equilibrium spacing Δs=2.325 and identical masses,yielding mass per area ρsb.

Fig.2 a LSM triangular lattice of flexible plate.Open squares indicate nodes at the boundary that are restricted in the vertical direction.Darker and lighter circles indicate mass nodes in the active and passive portions,respectively.b Distribution of external forces on single bending unit to model a local couple at the center node

In order to model an internal bending moment within the LSM framework,we apply forces judiciously to bending units parallel to the x-direction within the active section.Figure 2b illustrates the appropriate distribution of the forces on a single bending unit.As shown,a bending unit is composed of three nodes,i−1,i0,and i+1.The distance between nodes is given by r±1,0=|r±1,0|,where r±1,0=r±1−r0is the difference between the nodal position vectors.We apply the bending forces to i+1and i−1in the direction perpendicular to both r±1,0andˆy,which is the direction of the imposed internal moment(into the page).We emphasize that the bending units concerned are only the ones parallel to the x-direction. The force directions can be computed by the following

Using this procedure,a positive bending moment will lead to positive curvature in the x z-plane,presupposing that the entire bending unit does not rotate entirely around so that node i−1and i+1have switched positions.The bending nodal forces are thus given by

Here,Nr=11 is the number of horizontal rows in the lattice. The opposite force on node i0ensures that the total bending force on a bending unit is zero,representing a local couple at i0.These bending forces are applied to all bending units within the active region.

The mass nodes experience forces due to the springs,the fluid at the interface,and forces due to the applied moment for active nodes.The displacement of the mass nodes is computed by integrating Newton’s equations of motion using the velocity Verlet algorithm[69,70].Forevery single LBM time step,we perform the LSM integration eight times to ensure numerical stability.

The computational fluid domain size is 16 L×6 L×8 L with grid spacing Δc=2.In order to capture more accurately the fluid flow,the grid is refined near the plate surface. This refined grid has size 4 L×2 L×3 L and spacing Δf=1 is appropriately coupled with the coarser grid at their boundaries[71].On the outer boundaries of the coarse grid,periodic boundary conditions are applied,modeling an unbounded fluid.

The solid nodes are independent from the fluid lattice nodes and form a boundary surface that moves within the fluid domain.Our fluid–solid coupling procedure has been used and extensively tested in previous studies[52,53,72–80].Briefly,a local set of three solid nodes define a triangular surface adjacent to fluid nodes.Distribution functions from these adjacent LBM nodes propagate onto the surface and reflect,transferring momentum.This momentum transfer is associated with fluid forces that act on the three surface nodes.At the same time,an interpolated bounce-back condition at the LSM surface is applied in the LBM distribution functions,leading to no-slip and no-penetration boundary conditions[81].Details of the fluid–solid coupling procedure are described in a prior study[76].

To simulate free locomotion,we impose a uniform flow throughout the fluid domain and let the plate translate freely in the x-direction.The initial flow velocity is chosen so that free swimming velocity relative to the fluid domain is nearly zero so that the plate stays within the refined grid.The large computational domain also ensures that the far field velocity does not change appreciably as the plate moves forward.The simulations are performed for50 oscillation periods to ensure that the plate locomotion reaches a periodic steady state.

3 Model validation

In order to validate our LSM model,we calculate the static deflection of the internally actuated plate under the application of a constant moment.We consider a cantilevered plate with active and passive sections.In this case the plate bending deformation is in one dimension and can be modeled by the moment–curvature relationship for beam bending.Mathematically,the curvature κ(s)along the plate length in terms of the arc length coordinate s is expressed as

In these expressions,M is a constant moment that is applied on the active section for a length d and EI is the one-dimensional bending rigidity.The equation states that the curvature is constant within the active section,while it is zero in the passive section—the deformation extends in a straight line.The plate deformation curve z(x)can be expressed parametrically as follows

Here,R=EI/M is the radius of curvature.

Fig.3 Static deflection of an internally actuated plate with active and passive sections.Symbols represent LSM simulations,while lines represent the analytical solution good agreement is found between simulations and theory

We use our LSM with internal actuation model as described in the previous section to compute the deformation for R/L=9.6 and d/L values of 0.2,0.6,and 1.These bending curves are shown in Fig.3 with symbols.As shown, the plate maintains constant curvature with its slope increasing until the active–passive boundary at x=x(d)≈d.In the passive section,the plate deformation continues in a straight line until its free end.Consequently,the plate with the largest active section exhibits the largest deflection.

Our results are compared to the analytical solution, depicted using the straight and dotted lines as shown in Fig.3. We find good agreement between the LSM results and the analytical solution,differing by no more than 2%,which is related to the grid resolution of our LSM model.Therefore, the LSM moment actuation model accurately captures the beam bending behavior.

The LBM model and the coupling procedure between LBM and LSM has been previously validated and used for studies in flexible flapping aerodynamic and free swimming of plunging elastic plates[52,53,72].To examine the effect of grid density on the solution accuracy,we have performed a grid convergence study.We found that doubling the grid size did not change our results by more than 1%.

4 Results and discussion

As previously discussed in the introduction,we investigate the free locomotion of sectioned elastic plates that are internally actuated in the section closer to the leading edge.The goal of the study is to understand how the swimming performance is affected by the size ratio between the active and passive sections while keeping total length of the plate constant.In addition,we investigate the bending patterns that arise from varying its bending rigidity.We focus on the regime near the first natural frequency where resonance amplification results in the larger deflections and faster swimming.

We first identify the parameters that characterize the system and list the corresponding values used in our simulations.All dimensional quantities are given in LBM units. The plate dimensions are w=20 and L=50,and the plate is actuated at a constant driving frequency with period τ=2π/ω=2000.This defines the characteristic velocity Uc=L/τ,which represents the number of body lengths per period.We set the Reynolds number to be a constant in our simulations R e=ρUcL/μ=400,where ρ=1 is the fluid density.This determines the dynamic viscosity μ in the simulations.In addition,the mass ratio χ=ρL/(ρsb)is set to 2.5.The mass ratio characterizes the ratio between added and apparent masses,and this value is appropriate for artificial aquatic robots that propel using oscillating flexible plates. The mass per area ρsb thus can be determined.Moreover, the reduced moment

The dimensionless parameter characterizing bending rigidity is treated specially.An oscillating flexible plate in fluid has a natural frequency ω1,fthat depends on bending rigidity EI.Consequently,when the bending rigidity is varied,so does the natural frequency.We define the frequency ratio φ=ω/ω1,fto be the ratio between the driving frequency and first natural frequency in a fluid[82],where ω1,fis computed using linear theory for thin,large aspect ratio plates using the following expression

Here,λ1is the smallest root satisfying 1+cosλcoshλ=0 andΓ is the hydrodynamic function,whose values depend on aspect ratio and recursively on ω1,f.The values of Γ can be found elsewhere[82].Our previous study has shown that for similar plate parameters ω1,festimates the natural frequency in fluid to within 2%[53].The use of the frequency ratio allows us to characterize the current oscillation state by its proximity to resonance.

For plates with different values of d/L,we sample the range of φ between 0.8 and 2.0 and compute the period-averaged velocity U after the swimmer reaches a steady state. We plot our results of U/Ucversus φ in Fig.4,where different lines indicate different values of d/L.We find that the swimming velocity is maximized in the vicinity of the first natural frequency for each case of d/L.Indeed,the resonance oscillations near the first natural frequency are linked to faster swimming[53].Furthermore,the maximum swimming velocity is found to be roughly the same for all cases, except for that of d/L=0.2.This is surprising,as the fully actuated plate is not the fastest and even slightly slower than the plate with d/L=0.6.At a higher frequency ratio ofφ=2.0,the discrepancy is even more pronounced with the velocity at d/L=0.6 several times faster than that of d/L=1.At higher frequency ratios,having a passive section appears to dramatically enhance the swimming performance. We conclude that at all post-resonance frequencies,the plates with passive attachments(of size ratios d/L=0.4–0.6) achieve either equal or better performance than a fully actuated plate with the same total length.

Fig.4 Dimensionless swimming velocity as a function of frequency ratio velocity is maximized near the first natural frequency for all active section sizes.At higher frequencies,the plate with either d/L=0.4 or 0.6 achieves the fastest swimming speeds

To identify the physical mechanism that allows the passive flap to enhance the swimming velocity,we first examine the trailing edge deflection amplitude,δt,0.A larger trailing edge deflection is usually correlated to faster swimming [52,53].In Fig.5a,we plot δt,0/L as a function of frequency ratio φ.The data shows that,similar to swimming velocity, δt,0/L is also maximized near the first natural frequency,and the deflection amplitudes are nearly the same other than a lower amplitude for d/L=0.2.The similar trends between trailing edge deflection and velocity in the resonance regime suggest that a larger deflection leads to faster swimming. This does not hold,however,at higher frequencies when the trailing edge displacement is still nearly the same,but the plates with passive attachments are significantly faster than the fully actuated one.

To explore the correlation between trailing edge deflection and swimming velocity,we graph the velocity as a function of deflection in Fig.5b.We find that in general,the larger the trailing edge deflection,the larger the swimming velocity.For plates with spatially similar bending patterns,one with larger trailing edge amplitude would create a larger momentum flux behind the swimmer and,thus,generate more thrust.

Fig.5 a Trailing edge deflection as a function of frequency.Despite higher velocity at higher frequencies for d/L≥ 0.4,trailing edge deflection remains nearly the same for these swimmers and fully actuated swimmer with d/L=1.b Velocity versus trailing edge deflection showing that in general a larger trailing edge deflection implies faster swimming

This explains the poorer performance of the d/L=0.2 plate.The moment is applied near the leading edge,far away from the trailing edge,and as such is unable to oscillate the larger passive portion with sufficiently large amplitude compared to plates with larger active sections.Despite the general correlation of larger displacement with faster swimming,Fig.5b shows that for the same trailing edge amplitude, the velocity can also vary significantly and can change up to three times.The slower velocities correspond to fully active plates at higher frequency ratios.These observations suggest that the addition of a passive section affects the bending patterns of the fin which significantly alters the swimming performance.

In order to characterize the differences in the bending patterns,we examine two cases:d/L=0.6 and 1,both at the frequency ratioφ=2.The maximum trailing edge displacement differs by no more than 20%,but surprisingly,the fully actuated plate is nearly five times slower.In Fig.6a,we plot the time evolution during one period of the non-dimensional internal moment M′(t),trailing edge deflection,δt(t)/L, and trailing edge angle,αt(t)/L.We find that the deflection curves(the solid lines with symbols)for both cases are in phase,which is expected because the frequency ratio is the same.The deflection amplitudes are confirmed to be about 20%apart.Also,the deflection curves are nearly out of phase with the applied moment,which again is expected because the frequency ratio is higher than the natural frequency.

Fig.6 a Time history of trailing edge kinematics for plates with d/L= 1 and 0.6 at φ=2.Note that the trailing edge deflection and angle are out of phase for d/L=1.Snapshots of deflection curve are shown for b d/L=1 and c d/L=0.6.The negative slope at the trailing edge for d/L=1 contributes to loss of swimming performance

The most pronounced differences between the two cases are in the trailing edge angle curves.For d/L=0.6,the faster plate,the trailing edge angle(dotted line)is found to be in phase with the corresponding deflection.In contrast, the trailing edge angle for d/L=1,the poorer performer, is out of phase with the deflection.The bending patterns are visualized in Fig.6b for d/L=1 and Fig.6c for d/L=0.6 as snapshots of the instantaneous deflection curves during one period.In Fig.6b,the trailing edge angle and deflection are out of phase,so when the deflection is maximized,the angle is negative.The result is a shape that“cups”the fluid, limiting the momentum flux that leads to forward thrust.In contrast,Fig.6c depicts the bending pattern for a plate with a passive attachment.Here,the angle and deflection are in phase,so at the maximum extent of the trailing edge,the angle is positive,leading to a slope that effectively pushes fluid backwards to enhance thrust.

The physical mechanism leading to these bending patterns can be explained by analyzing the actuation mechanism.An internal moment acts to change the local curvature,which in turn affects the slope of the deflection curve.When the plate is fully actuated,the trailing edge exhibits a significant local curvature because it lies within the actuation region.Thus, we expect that the angle at the trailing edge responds quickly to the applied moment,i.e.,the angle and moment have a small phase lag.Indeed,a smaller phase lag is seen in Fig.6a between the moment and angle time histories for d/L=1 compared to d/L=0.6.Thus,for fully actuated plates,at the time of maximum deflection,the internal moment,which is out of phase with deflection,creates the negative trailing edge slope.In turn,this leads to a loss of swimming performance.

Fig.7 Phase between trailing edge deflection and angle as a function of frequency ratio.The passive flap causes the phase to stay nearly zero in a wide frequency range for d/L=0.6 and smaller,which contributes to better swimming performance

The aforementioned effect does not occur with the passive attachment because the internal moment is confined to the active section and does not directly affect the curvature of the passive attachment if the passive section is long enough.Furthermore,this trailing edge curving effect does not occur with lower frequency ratios closer to the natural frequency because the moment,deflection,and angle are all in phase regardless of the size of the active section.Thus,the swimming performance at low frequencies is all rather similar among all swimmers(with the exception of d/L=0.2 whose poor performance is related to small deflection).

To corroborate further our analysis,we plot in Fig.7 the phase Δθ between the trailing edge deflection and angle for multiple simulations.For active sections sized d/L=0.6 and smaller,the deflection and angle stay nearly in phase for all frequency ratios.This bending pattern is correlated with better performance.However,for d/L=0.8 and 1,phase increases with frequency ratio.This suggests that a too small passive attachment allows the internal moment to affect the trailing edge curvature and change the trailing edge angle detrimentally.

Thus,we conclude that the addition of the passive attachment leads to a better swimming performance at higher frequencies because it suppresses the detrimental curvature at the trailing edge.At lower frequencies,the passive attachment does not affect the swimming performance unless it is too large,in which case the reduced trailing edge deflection suppresses the generation of adequate thrust.

The previous discussion was centered on the swimming speed.Another important parameter for characterizing the swimmer performance is the power consumption.The total input power is calculated in our LSM model as the dot product between the external nodal forces FM,iand the nodal velocity vi,summed over all nodes:

Fig.8 a Power coefficient and b swimming economy as a function of frequency ratio.With decreasing d/L,power decreases while swimming economy increases.Plates with passive flaps exhibit better swimming economy as well as faster velocity

Here,the brackets indicate period-averaging.The power is normalized by characteristic powerw L to yield the power coefficient CP=P/Pc.In Fig.8a,we plot the characteristic power as a function of frequency ratio for the different values of d/L.We find that the input power is maximized near the first natural frequency,similar to the trends found for the swimming velocity(Fig.4)and the trailing edge deflection(Fig.5a).We also find that the power increases with increasing d/L,which is expected since the deflection in the active section(and by extension vi)is smaller when the active section is smaller.

To quantify how efficient swimming is,we introduce the swimming economy ε=(U/Uc)/CP,defined as the ratio between dimensionless swimming velocity and power coefficient.This quantity represents the distance traveled by the plate per unit applied work.In Fig.8b,we plot ε as a function of frequency ratio and find that the swimming economy decreases with increasing d/L.Thus,the swimming economy is the best with the smallest active section,but in this case the swimming velocity is slower.To swim fast at a wide range of frequencies,the swimmer should have an active section equal to about a half of its full length,in which case the swimming is more economical than the fully active plate. Thus,an addition of the passive flap is advantageous for both swimming speed and economy.

Fig.9 Swimming economy as a function of center of mass deflection of internally actuated swimmers.Swimming economy generally decreases with increasing center of mass deflection with the exception of the poorer cases corresponding d/L=1 and 0.8 at high frequencies

We have previously showed for a plunging passive swimmer that a higher swimming economy is correlated with smaller center of mass displacement[53].Large vertical deflections in the bending pattern lead to strong side vortices and contribute to vortex induced drag,hindering the swimming efficiency[52,63].Here,we plot the swimming economy of internally actuated swimmers as a function of the center of mass deflection in Fig.9 and find a general trend in which a smaller center of mass displacement leads to higher swimming economy.

Interestingly,we find for the cases showing poor economy at higher frequencies corresponding to d/L=1 and 0.8 that the economy slightly increases with increasing center of mass deflection.This is likely because the detrimental trailing edge angle dominates the swimming performance, whereas the increased viscous losses due to a greater center of mass displacement have a secondary effect.In this case,a reduction of the trailing edge angle can lead to slightly better economy even for an increased center of mass displacement.

5 Summary

MFC bimorphs and other active materials are attractive oscillating propulsor alternatives to traditional cumbersome motor-based actuators[6–11]and have been used recently in untethered aquatic devices[39].Our study demonstrates that the addition of passive flaps enhances the swimming speed and efficiency of moment actuated flexible fin propulsors. This conclusion agrees well with recent experimental observations that the addition of a passive flap to active fins leads to increased swimmer thrust and propulsion speed[41].

Specifically,we used three-dimensional,fully coupled computer simulations to investigate free underwater locomotion of an internally actuated elastic plate with a passive elastic attachment.The internal sinusoidal moment applied to the swimmer modeled piezoelectric actuation of an MFCbimorph.We isolated the hydrodynamics of the plate representing a swimmer fin and tested the swimming performance with different active to passive section size ratios.This allowed us to determine the effect of a passive attachment while holding the total length of the plate constant.

Our results indicated that the addition of a passive elastic flap improved both swimming speed and swimming efficiency.By probing the trailing edge kinematics and deflection curves,we determined that if the active and passive sections are of similar size,the overall performance is optimal.A large passive attachment was found to hinder overall swimmer deflection leading to slow swimming velocity.When the active section was too large,the internal moment actuation led to swimmer profiles that are not conducive to generating thrust at higher(post-resonance)frequency ratios.Specifically,the internal moment changed the angle of the trailing edge so that it is out of phase with the deflection.

We characterized efficiency using the swimming economy,and found that the economy increases with the size of the passive attachment.The plate with the largest passive section was the most economical,but also the slowest.For active section sizes in the range d/L=0.4 and 0.6(i.e.,the active and passive sections are similar sized),the swimming velocity was overall the fastest and efficiency was better compared to fully active swimmers.In other words,we showed that by simply adding a passive attachment of similar material properties to the smart plate actuator better swimming performance can be achieved.We correlated this effect with the changes in the swimmer bending pattern.Thus,our results will be useful for the design of faster and more efficient biomimetic underwater propulsion devices.

Acknowledgments The authors would like to thank L.Cen and A. Erturk for the stimulating and insightful discussions.

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✉ Alexander Alexeev alexander.alexeev@me.gatech.edu http://cfms.gatech.edu

1George W.Woodruff School of Mechanical Engineering, Georgia Institute of Technology,Atlanta,USA

7 December 2015/Revised:4 March 2016/Accepted:6 May 2016/Published online:7 September 2016