DESIGN OF A HIGH EFFCIENCY, LARGE STROKE, ELECTROMECHANICAL ACTUATOR

Eric F. Prechtl: Graduate Research Assistant, Massachusetts Institute of Technology, 77 Massachusetts Ave Cambridge, MA 02139-4307 USA, Office: 37-391, Email: prechtl@mit.edu , Telephone: 617-253-3267, Fax: 617-258-5940

Steven R. Hall: Associate Professor of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Ave Cambridge, MA 02139-4307 USA, Office: 33-207, Email: srhall@mit.edu , Telephone: 617-253-0869, Fax: 617-258-7566

Keywords: Efficiency, Mass, Actuator, Bandwidth, Piezoelectric, Servo-flap, Helicopter, Rotor

ABSTRACT

Large stroke, electromechanical actuator designs are considered. Special emphasis is placed on actuators designed to power a trailing edge servo-flap system for feedback control of helicopter rotor vibration, acoustics, and aerodynamic performance. A survey was conducted comparing the advantages and disadvantages of a number of actuator designs. The major conclusions from this survey indicate that any successful actuator design will utilize a high bandwidth active material, produce large amplification of the active material stroke, and incorporate a simple compressive pre-stress mechanism, while remaining efficient in a mass normalized sense. The mass efficiency, defined as the ratio of the specific work performed by the actuator to the specific energy available in the active material element, was used as a metric to rate the actuators considered in the survey. This metric is appropriate in aerospace applications where weight is critical. The most feasible discrete actuators are those where the active material reacts against an inert support frame housing. An upper bound on the mass efficiency of this type of actuator is shown to be a function of the ratio of active material to frame specific modulus. A new high efficiency discrete actuator, the X-Frame Actuator, is described. A prototype of this actuator was built and tested to confirm the predicted performance. The prototype demonstrates an output energy density of 14.6 ft-lb/slug. It has a bandwidth of about 540 Hz when driving a nearly impedance matched load.

1 INTRODUCTION

There exist many applications which require the use of fast acting, large stroke actuators. Active materials, such as piezoelectric ceramics, are likely choices to power such devices, due to their inherent high bandwidth. Typically, active materials produce large forces and small displacements, so significant amplification is usually required when using these materials in actuator applications.

In this paper, we consider an actuator developed for a specific application, namely, the control of a helicopter rotorblade trailing edge servo-flap to improve rotor vibration, acoustics and aerodynamic performance. Previous studies have shown that the use of blade-mounted actuators to control a servo-flap offers a number of distinct advantages. For example, Hall et al. showed that blade actuators can be used to reduce rotor induced power losses.¹ Other studies have also shown that rotating frame actuators may be used for higher harmonic control (HHC) with much lower power requirements than traditional techniques.^2,3

Active materials are well suited for rotor control because of three main reasons. First, active materials, such as piezoelectric ceramics, have the required energy density to perform the rotor control objective. Second, active materials can actuate over the bandwidth required for rotor control. Third, the electro-mechanical nature of active materials allows them to be controlled by electrical signals which are easily transferred into the rotating frame through a standard slip-ring. Other actuation technologies such as hydraulic or pneumatic actuators require special slip-rings to deliver the control signals and would encounter chronic maintenance problems due to the severe operating environment in the high centrifugal field of the rotor system.

Spangler and Hall first proposed trailing edge servo-flap actuation using a piezoelectric bender actuator.^4,6 The bender was used because it was an expedient method to amplify the active material stroke. Spangler and Hall demonstrated the feasibility of this actuation method through wind tunnel testing of an airfoil typical section. While they obtained appreciable flap deflections and force authority (±6 deg at 23.8 m/s test velocity), they found their design did not work entirely as expected due to hinge friction and backlash. Later work improved on this actuation concept by implementing a tapered bender and a flexural connection between the bender and the flap.⁷ Chopra et al. adopted the bender concept and have performed a number of actuator studies on Froude scaled rotor models.^8,9

The limitation of the traditional bender actuator is that only a small part of the active material strain energy can be utilized because its operation relies on the indirect, “31", active material effect. In principal, more energy is available when the direct, “33", active material effect is used, e.g., as with piezoelectric stack actuators. However, to achieve the required stroke, mechanisms must be used to amplify the small strains. Bothwell et al. used an extension-torsion coupling mechanism to amplify longitudinal motion to produce rotary motion.¹⁰ Fenn et al. have also proposed a discrete actuator that uses two magnetostrictive expansive elements oriented at a shallow angle for geometric amplification. Large stroke actuators based on amplifying active element stroke are also available commercially, e.g., Physik Instrumente sells such a large stroke actuator.¹¹ The performance of each of these actuators is considered in Section 4. Other researchers have developed actuators not specifically addressed in this paper. For example, Giurgiutiu et al. have built an actuator using hydraulics to amplify active material motion.¹² Jänker et al. are developing two new actuator concepts, a planar Disc Actuator and a Hybrid Actuator using a piezoelectric stack and an inert frame amplification mechanism.¹³ And, finally, Straub et al. are developing an actuator that is also designed for helicopter rotor control utilizing a mechanical amplification of the active material stroke.¹⁴

In this paper, we describe the development of a new high-efficiency actuator. The actuator was developed for use in active rotor control, but may be applied to other applications as well. In Section 2, we describe the requirements for the actuator. In Section 3, we give a brief survey of active materials, describing the metrics used to compare the effectiveness of the different materials. Section 4 is a description of design considerations, which apply to any active material actuator where weight is an issue. In particular, we develop theoretical bounds on the achievable mass efficiency of active material actuators, and compare several existing designs to these bounds. Based on these analyses, we offer several actuator design guidelines. Based on these guidelines, we present in Section 5, a new design, the X-Frame Actuator, which has performance significantly better than existing designs. Finally, Section 6, presents experimental results on the X-Frame Actuator prototype.

2 DISCRETE ACTUATOR REQUIREMENTS AND DESIGN METRICS

In support of this study, Boeing Helicopters, through the use of their TECH-01 rotor analysis tool, identified a number of requirements that any actuator for servo-flap control must satisfy. These requirements are collected here:

Force: The actuator must be able to react operational hinge moments.

Stroke: The actuator must be capable of ±5 deg of flap motion.

Bandwidth: The actuator must have a bandwidth appropriate for higher harmonic control (> 4/rev).

Mass: The actuator should be light, with the actuation adding less than 20% to the blade weight.

Integration: The actuator must fit within the blade spar for acceptable mass balance.

Lifetime: The actuator fatigue life must exceed 200,000,000 cycles.

Environment: The actuator must be able to perform in the operational load, vibration, and temperature environment.

The material used to power a discrete actuator is the limiting factor for most of these requirements. An in-depth discussion of the active material properties in relation to these requirements is given in Section 3.

Independent of the active material is the amplification mechanism. Because of the large centrifugal field in the rotorblade environment, the mass requirement is very important. The mechanism's efficiency in transforming internal energy into usable work determines the required mass of the device.

In general, the amplification mechanism consists of an active element (or active elements), which provides the actuator force and displacement, and a support structure, which reacts the loads. Compliance in the support structure leads to mechanical inefficiencies in the actuator. The impact of frame compliance on actuator performance is measured by the mechanical efficiency of the actuator, defined as:

(1)

which is the ratio of the actuation output energy to the active element energy. Here, K_a is the stiffness measured at the actuator output, q_f is the free (unloaded) displacement corresponding to the induced strain, , in the active element. E_e and V_e are the Young's modulus and volume of the active material element, respectively.

A straightforward way to increase the mechanical efficiency is to incorporate a very stiff frame. But, such a frame would also be very massive. In applications where weight is important, it is preferable to sacrifice some mechanical efficiency in order to minimize weight. This tradeoff can be quantified by the mass efficiency of the actuator, defined as:

(2)

Augmenting the mechanical efficiency with the ratio of active element mass, M_e, to total mass, M_tot, makes the mass efficiency a useful design metric reflecting the trade between frame compliance and frame mass. The mass efficiency can be thought of as the ratio of the specific work delivered by the actuator to the specific energy available in the active element.

Combining the required energy for a given application with the actuator mass efficiency and active element energy density gives an accurate estimate of the required actuation system mass. Optimum actuator design is largely focused on maximizing the product of actuator mass efficiency and active element energy density.

3 MATERIAL SURVEY

A number of active materials were considered for the actuator application. In comparing the various materials, certain criterion were used. Each of these is described here, particularly with respect to its impact on actuator design.

Energy Density. The energy density is the specific strain energy an active material can deliver. It is defined as:

(3)

where is the density of the active material element. The product of the energy density of a material and the mass efficiency of the actuation mechanism gives the specific work a particular actuation/material combination can perform. Thus, for a given actuation mechanism configuration to perform a certain amount of work, use of a larger energy density material implies a lighter actuator.

Maximum Strain. A large induced strain is desired because it is directly related to the energy density, through Equation (3). More importantly, a large induced strain reduces the required stroke amplification of the discrete actuator. Amplifications on the order of 20:1 are feasible, but efficient amplifications greater than that are difficult to obtain.

In this paper strains are reported as peak-to-peak (PP) values so that materials with induced strains that are linearly dependent on the applied field, like piezoelectric ceramics, can be compared directly against materials with induced strains that are quadratically dependent on the applied field, like magnetostrictive alloys. Maximum strains were estimated by looking at the full, non-linear experimental curves of strain versus applied field. The maximum usable strain was taken as the range of strain without noticeable material saturation. Note that these maximum strains are consistently higher than those specified by most vendors, see e.g., Giurgiutiu et al.¹⁵

Bandwidth. The frequency range over which the actuator will be used defines the required bandwidth. The rotor control objective requires a bandwidth greater than 4/rev. This criterion eliminated some high energy density active materials, like shape memory alloys from consideration.

Longevity. Material lifetime is an important issue because many active materials are inherently brittle ceramics. The concern over the longevity of certain active material systems eliminated them from consideration. For example, the choice between using plate-through or co-fired stacks was based on longevity concerns, as discussed below.

Technical Maturity. Technical maturity is a measure of the available knowledge or previous experience that exists with a material. The technical maturity of a material system must be seriously considered in terms of its potential risk before including it in an application.

Linearity. All active materials exhibit some nonlinearity. One common example is the nonlinear strain behavior of piezoelectric ceramics. A highly nonlinear material is difficult to integrate into a linear control system.

Temperature Sensitivity. Materials with low temperature sensitivity are desired due to the wide temperature range in which helicopters operate. Unfortunately, most active materials have some temperature dependence.

Cost. The use of expensive materials must be weighed against its potential benefit.

Four general material types were considered for the present application; piezoelectric ceramics, magnetostrictive (MS) alloys, shape memory (SM) materials, and electrostrictive ceramics. With the exception of magnetostrictive alloys, all of the materials can operate using the strain induced parallel to or transverse to the applied field. These two operational modes are often referred to as the 33 or 31 effect, respectively. Generally, the parallel (or longitudinal) strain is used in stack actuators while the transverse strain is used in planar configurations. In addition to these traditional actuation modes, Active Fiber Composites (AFC) were also considered as a planar actuation alternative.¹⁶

Preliminary material comparison eliminated shape memory alloys and shape memory ceramics from consideration. Shape memory materials boast great energy density levels but have limited bandwidth. Shape memory ceramics have improved bandwidth characteristics but were discarded because they are a relatively immature technology.

The main functional difference between electrostrictive and piezoelectric ceramics is that the induced strain in electrostrictive ceramics is a quadratic function of the applied field, whereas the induced strain in piezoelectric ceramics is a linear function of the applied field. Thus, the induced strain in electrostrictive ceramics is independent of the field polarity. A material that exhibits both electrostrictive and piezoelectric properties is lead magnesium

niobate - lead titanate (PMN-PT). The relative amount of PMN and PT determines whether the material exhibits a dominant electrostrictive or piezoelectric character. In fact, both the electrostrictive and piezoelectric materials (longitudinal actuation only) considered in this study are PMN-PT type materials, the difference being their relative amounts of PMN and PT. The data used for the electrostrictive materials was taken from Fripp.¹⁷ The experimental performance of these materials showed very low energy densities as well as strong electric field dependent and temperature dependent nonlinearities. For these reasons, this particular electrostrictive material was also excluded from consideration.

Commercially available piezoelectric ceramics and magnetostrictive alloys have comparable nonlinearities, durabilities, as well as large voltage and current levels, respectively. Active fiber composites (AFCs) share many of these properties, but are more robust to tensile loads. Active fiber composites, however, have the drawback of being a more expensive and immature technology than commercially available piezoelectric and magnetostrictive materials.

The energy density became the criterion used to differentiate between the active material systems. The energy density comparison for these active materials is presented in Table 1. The Young's modulus and density reported in this table were obtained from available literature on the materials. The peak-to-peak strain for the piezoelectric ceramics was estimated from the data in references 18 and 19. The maximum magnetostrictive strain was estimated from data published by ETREMA.²⁰

Material	Maximum PP Induced Strain, microstrain	Young's Modulus Ee 10^6psi	Density re (slug/ft^3)	PP Energy Density Ue (ft-lb/slug)
Bulk EC-98 (33)	1650	6.9	15.2	88
EC-98 Stack (33)	1650	4.8	varies	52
Magnetostrictive	1500	4.3	17.9	39
PZT-5H (31)	700	8.8	14.6	21
AFC	1150	4.8	9.3	49

Table 1. Typical linear active material properties

The energy density values in Table 1 vary from those reported by vendors (see e.g., Giurgiutiu et al.¹⁵ -Note that the energy densities reported here correspond to the “output energy per active material mass" of Giurgiutiu et al. times a factor of four, because Giurgiutiu includes an “impedance matching" factor of 1/4 in his calculation of maximum output energy.) because of the assumed strain ranges, as discussed above. Later, in Section 6, experimental data is presented to support the strain assumption reported in Table 1 for EC-98 stacks.

The table is split between longitudinal (33) and planar (31, AFC) actuation materials. In the longitudinal section, the properties for bulk piezoelectric and magnetostrictive materials are presented. In practice, it is difficult to attain the energy density of the bulk material. For example, magnetostrictive materials require a heavy solenoid to actuate the material. Accounting for the extra mass of a typical solenoid would substantially reduce the effective energy density of the material. Similarly, piezoelectric stacks have lower energy density than the bulk material mostly because of compliance losses in the bond layers present between the stack layers. The extra mass of end-caps and the electrode bus also lower the effective energy density of the material. EC-98 stacks supplied by the EDO Corporation are used to power the X-Frame discrete actuator. Assuming a cut down in the effective stack stiffness of 0.7 ±15% due to bond layer compliance,²¹ and taking into account the mass of endcaps and the electrode bus, the energy density of the EC-98 stacks is estimated as shown in Table 1.

The only stack design considered in this study was a plate through type, i.e., where the electrode between each layer covers the entire cross-section of the stack, as shown in Figure 1b. However, a different manufacturing technique for piezoelectric stacks uses a co-firing process. Here the electrodes and ceramic material are processed together. This technique yields stacks with much smaller compliance losses between stack layers. Also, because the wafer thickness in these stacks is typically thinner than plate-through designs, much lower voltage levels are required to create a certain electric field. However, the electrodes in co-fired stacks normally extend only partially through the ceramic, as shown in Figure 1a. Termination of the electrode within the ceramic leads to a stress concentration at the electrode tip and can limit the lifetimes of these stacks. Plate-through stacks, as shown in Figure 1b, are used for the rotor control objective because this design presents lesser risk of failure than co-fired stack designs. The disadvantage in using a plate-through stack design is that extra bond layer compliance will limit the stiffness and blocked force capability of the actuator.

Even with the bond layer losses and additional mass associated with commercially available stacks, their energy density exceeds that of bulk magnetostrictive materials. Thus, piezoelectric stacks are the preferred longitudinal actuation system.

Because the mass efficiency of a planar actuator can differ from that of a longitudinal actuator, the product of the achievable mass efficiency of the mechanism and the energy density of the associated material must be used when comparing longitudinal and planar actuation materials. This product represents the specific work such an actuator can perform. While it is possible to come up with planar actuators utilizing PZT-5H material with relatively high

mass-efficiencies, e.g., the bender actuator,⁷ such an actuator would have to be 2.5 times more mass efficient than a stack actuator to be viable. Of course, active fiber composites are a more attractive alternative in this sense, because of their high energy density. Use of AFCs in a bender actuator is discussed further in Section 4.1.

4 ACTUATOR DESIGN CONSIDERATIONS

One drawback to using high bandwidth active materials is that they typically have very small displacements, albeit with high force. Thus, a stroke amplification mechanism must be employed to use these materials for servo-flap control or other applications. The mechanism used to achieve this amplification, the supporting frame components, and the active material define the actuator. This section presents the design and analysis of a number of different actuator concepts. The section concludes by highlighting a number of important design axioms that lead to successful

actuator designs.

4.1 The piezoelectric bender actuator

One method of amplifying small strains is by using a bending actuator.^4-7 Spangler and Hall and Hall and Prechtl developed this actuation concept, as shown in Figure 2.

The bender is connected to the trailing edge flap through a mechanism incorporating three flexures. Because the material reacts against itself, the mass efficiency and mechanical efficiency are very high. The nominal mass efficiency of a rectangular cross-section bender is 56%. By tapering the bender, the mass efficiency can be increased to 75%. An experimental bench test of this bender has been performed at MIT demonstrating efficiencies greater than 60%.⁷

The high mass efficiency of the bender actuator is mitigated by the fact that the transverse energy density of piezoelectric ceramics is 2.5 times smaller than the longitudinal energy density. Even with high mass efficiencies, benders constructed with monolithic ceramics have output energy densities too low to be competitive with stack actuator designs.

Active fiber composites, on the other hand, yield energy densities approaching longitudinal energy densities. A bender constructed of AFCs should yield output energy densities much greater than a monolithic bender and comparable to existing stack actuator designs. Such an AFC bender was constructed at MIT.^22,23 Experimental results demonstrated energy densities on the same order as the monolithic piezoelectric ceramic bender previously tested by Hall and Prechtl.⁷ The reason for the lower than expected performance may have been compressive depolarization of the AFC fibers.

The bender actuator also suffers from a number of other problems. One of its main problems for rotor control is the mass of the actuator is behind the 1/4 chord of the airfoil, which can excite aeroelastic instabilities.²⁴ Leading edge weights can be added to maintain a sectional CG at the 1/4 chord but such weight would lower the overall mass efficiency of the device. Optimizing the properties of such a design results in a maximum mass efficiency of 37%. Another problem with the bender is that it must be mounted in the front of the airfoil and extend back to the trailing edge, requiring significant modification to the existing rotor blade spar.

It is likely that the experimental problems encountered in the AFC bender can be overcome, resulting in a relatively high energy density actuator. However, because of the problems discussed above and because of the immaturity of active fiber composite technology, the risk in using an AFC bender was too high for this project.

4.2 Coupling mechanisms

A possibility for amplifying the small motion available from stack elements is to use elastic structures that couple two types of motion. For example, Bothwell et al.¹⁰ proposed using a composite tube with an anisotropic layup, which induces extension-twist coupling. Naturally, it is desirable to design the coupling mechanism to be as efficient as possible. In this section we derive the maximum possible mechanical efficiency of a coupling mechanism amplifier. As will be seen, the efficiency is quite low, unless the coupling is nearly perfect.

To begin, we hypothesize a generic coupling mechanism, with two generalized displacements (e.g., torsion and extension). These degrees of freedom are denoted by q₁ and q₂. Corresponding to the two degrees of freedom are generalized forces (e.g., torque and force), denoted by Q₁ and Q₂. The forces and displacements are related by

(4)

where K is a symmetric, positive semidefinite stiffness matrix.

Next, we assume that the coupling mechanism is connected to the load at the first degree of freedom (q₁), and an expansive element is connected to the coupling mechanism at the second degree of freedom (q₂). For example, a typical expansive element could be a piezoelectric stack. The expansive element characteristics are given by

Q_e = K_eq_e + Q_eb (5)

where Q_e is the force on the element, K_e is the element stiffness, q_e is the element displacement, and Q_eb is the blocked force capability of the element.

By compatibility and equilibrium,

q_e = q₂ (6)

Q_e = -Q₂ (7)

Then

Q₂ = -K_eq₂ - Q_eb = K₂₁q₁ + K₂₂q₂ (8)

Solving for q₂ in terms of q₁ and Q_eb, we have that

q₂ = - (K_e + K₂₂)^-1 K₂₁q₁ + (K_e + K₂₂)^-1 Q_eb (9)

Finally, we can determine Q₁ as

Q₁ =[K₁₁ - K₁₂(K_e + K₂₂)^-1 K₂₁]q1 + K₁₂(K_e + K₂₂)^-1 Q_eb (10)

= K_cq₁ + Q_cb (11)

where

K_c = K₁₁ - K₁₂(K_e + K₂₂)^-1 K₂₁

Q_cb = K₁₂(K_e + K₂₂)^-1 Q_eb

Note that Equation (11) is similar in form to the expansive element characteristic Equation (5). Indeed, Equation (11) may be viewed as the characteristic of the coupled actuator. The useful work that may be extracted from the element/coupling mechanism combination is

(12)

(13)

The energy available from the expansive element is

(14)

The coupling efficiency, h_c, is defined as the ratio of the work delivered at the output of the coupling mechanism to the available expansive element energy, so that,

(15)

This equation may be simplified somewhat by dividing both the numerator and denominator by . Then

(16)

where and are dimensionless parameters, given by

(17)

(18)

The parameter is the coupling coefficient, which describes how close the elastic coupling mechanism is to an ideal coupling mechanism. Since K is positive semidefinite, we have that

(19)

The parameter describes how stiff the expansive element is relative to the stiffness of the coupling mechanism.

When , the stiffness matrix K is singular, and the coupler acts as a flexible mechanism. That is, there is no inherent resistance to motion in the coupler. An example of such a device is a flexible lever on an ideal fulcrum. To get the most efficiency out of the coupled actuator one should make the expansive element stiffness low compared to that of the coupler, i.e., should be small. Indeed, when ,

(20)

Hence, the efficiency can be made arbitrarily close to unity by making arbitrarily small.

When , the coupler is no longer a mechanism, and there is resistance to motion, even when there is no load attached to q₁. Therefore, there will be a tradeoff between making small, which will produce little motion at all; and making large, which will result in significant motion, but will also waste energy as elastic deformation of the coupler. Therefore, there will be an optimal matching between the coupler and the expansive element. The optimum efficiency may be found by taking the derivative of h_c with respect to , holding fixed, and setting the result to zero. The optimal actuator stiffness is

(21)

For this matched condition the coupling efficiency is

(22)

Not surprisingly, when = ±1, the coupler is a mechanism, and we can achieve . Likewise, when = 0,the “coupler" has no coupling at all, and . For values of between 0 and 1, the efficiency likewise varies between 0 and 1. The surprising part is how close the coupler must be to a perfect mechanism to achieve reasonable efficiencies, as shown in Table 2.

Coupling parameter,	Optimum coupling efficiency,
0.5	0.0718
0.75	0.2038
0.9	0.3929
0.99	0.7527
0.999	0.9144
0.9999	0.9721

Table 2. Efficiencies for various coupling parameters

Our conclusion is that, with few exceptions, elastic coupling mechanisms are not feasible. Other researchers have built and tested actuators based on elastic coupling mechanisms. Bothwell et al. built and tested an [11]₂ Kevlar-Epoxy extension torsion tube designed according to a parametric ply angle optimization.¹⁰ The coupling parameter for this laminate is calculated using Classical Laminated Plate Theory as = 0:68, leading to a maximum theoretical mechanical efficiency of 15%. In practice, it is difficult to reach this optimum efficiency. For example, the data in 10 indicates h_c = 6:1%. Furthermore, once the extra mass needed to properly fixture and attach the actuators is taken into account it becomes clear that this is not a mass efficient actuation alternative.

4.3 Stack/inert frame actuators

The considerations in the previous section led us to investigate flexible mechanisms to amplify active element stroke. The fundamental issue to resolve in the design of such mechanisms is to identify those factors that limit their mass efficiency. This section presents the derivation of an upper bound for flexible mechanisms and presents the performance of two possible actuation mechanisms in light of this optimal mass efficiency argument.

4.3.1 Maximum achievable mass efficiency

Consider a simple model of a generic expansive element operating through a 100% efficient stroke amplifier, with amplification ratio a, and reacting against a support frame, as shown in Figure 3a. Now, assume the end
plate at the base of the element and the material supporting the amplification mechanism are ideal, i.e., infinitely stiff with zero mass. These ideal material regions are indicated by the shaded regions in Figure 3b. Therefore, we consider only the extension of the element and the frame material adjacent to the stack. Let the modulus, density, and cross-sectional areas of the expansive element and frame be denoted as E_e, E_f , r_e, r_f, A_e and A_f , respectively. Furthermore, assume the active material and stack have the same length, L. Note that this last assumption does not limit the scope of this analysis. The frame must be at least as long as the element; and unnecessarily long frames result in greater losses.

The output stiffness of this actuator is then given by

(23)

and the free deflection of the actuator is

(24)

The masses of the element and the total mass of the actuator is

M_e = A_eLr_e

M_tot = (A_fr_f + A_er_e)L

The mass efficiency of the actuator is

(25)

where the overline represents the element-to-frame ratio, e.g.,. Clearly, a tradeoff exists between making small, which will result in a light but overly flexible actuator and making large, resulting in a massive actuator with little compliance. There is an optimum ratio of frame to element cross-sectional area, found by taking the derivative of h_mass with respect to , holding and fixed, and setting the result to zero. This optimum area ratio is

(26)

For this optimum area ratio, the mass efficiency is

(27)

where a is defined as the ratio of active material to frame specific modulus, that is,

(28)

Note that for this optimum mass efficiency condition, the mechanical efficiency is

(29)

Equation (27) represents an upper bound on the achievable mass efficiency for a stack reacting against an inert frame. Figure 4a shows the optimum mass efficiency as a function of a. For example, EC-98 stacks reacting against a steel frame, has an a parameter of 0.164 and h_mass = 50:6%.

Ideal frame materials are easily found by examining plots of modulus versus density for common engineering materials, such as those given by Ashby.²⁵ Figure 4b is a modified version of such a material plot. It is possible to identify lines of constant mass efficiency and constant cross-sectional area in this type of plot by manipulating Equation (28) and Equation (26). These lines are shown in the figure labeled with optimum mass efficiency and cross-sectional area ratios for frames supporting EC-98 stacks. (Note that in Figure 4b, h_m stands for h_mass and stands for .) The lines of constant cross-sectional areas are important to consider when actuator size is an issue, such as fitting it within a rotor blade spar.

These material plots can be very effective tools for picking optimum frame materials. Examining the plot shows that exotic frame materials, such as diamond, boron (B) and silicon carbide (SiC) result in very high mass efficiencies.

However, other factors such as cost, longevity, or coefficients of thermal expansion may make certain frame materials unattractive.

4.3.2 Lever and Pyramid actuators

A number of stack/inert frame actuators were analyzed. The properties of each design were optimized to maximize the mass efficiency of the device. This maximum mass efficiency was compared to that predicted by Equation (27) to rate the performance of the amplification mechanism. This section discusses the conclusions drawn from studying two particular actuator designs.

A simple lever and fulcrum is an obvious method of amplifying stack motion by using an inert frame. Such actuators are available commercially. For example, Physik Instrumente sells such an actuator.¹¹ The operational

concept of the lever actuator is shown in Figure 5a. This actuator benefits from a simple amplification mechanism, easy incorporation of a pre-stress mechanism at the actuator output, and an ideal form factor for placing the actuator in tight spaces, such as a rotor blade spar for trailing edge flap actuation. However, the design requires that some loads are carried in bending, which is not very efficient. As a result, the mass efficiency is significantly lower than the theoretical bound. Numerical optimization of this actuation design, assuming perfect rolling contacts between an EC-98 stack and a steel frame, predicts a maximum achievable mass efficiency of 28%.

Another actuator, referred to in this paper as the pyramid actuator, is shown in Figure 5b. It consists of two stacks reacting against each other at a shallow angle, resulting in an amplified displacement. Such an actuator design was invented by Stahlhuth²⁷ and has also been proposed for the servo-flap actuation concept by Fenn et al.
Magnetostrictive actuators are used in this design, but piezoelectric stacks could easily be substituted. The stacks are supported by a titanium frame, with simple flexures providing rotational degrees of freedom at the stack ends.

Numerical optimization of this design showed that these flexures severely degrade the mass efficiency of the device. The source of this loss is due to the natural trade in the flexure design between axial stiffness and rotational compliance. An acceptable alternative to using flexures is to use rolling contacts between the stacks and frames. Such contacts do yield some Hertzian losses, but they are small in comparison to the losses associated with flexures.

Aside from the flexures, this actuator design has a high mass efficiency. Numerical optimization of this design, for EC-98 stacks reacting against steel frames, gives a maximum achievable mass efficiency of 36% for the design with flexures and 51% given perfect rolling contacts at the ends of the stacks. However, the main drawback of this design is that its amplification is dependent on the shallow angle of the stacks, which changes during operation, resulting in a nonlinear amplification mechanism. For designs yielding a nominal amplification of 15, operational strains can result in a 15% change in the amplification. Furthermore, such a stack arrangement could “snap-through" if, for example, the actuator encounters an impulsive load. In such a case the stacks would bifurcate to an equilibrium position that is a mirror image of that shown in Figure 5b.

4.4 Properties of an optimal discrete actuator

The survey of discrete actuators aided us in developing a number of axioms to use in designing an optimal discrete actuator. A successful actuator will usually incorporate many, if not all, of them. They are collected here:

Planar Actuators. Planar actuators, such as the piezoelectric bender, do not offer a significant energy advantage over optimally designed stack/inert frame actuators, have manufacturing difficulties and lead to poor sectional CG characteristics, as discussed in Section 4.1.

Coupling Mechanisms. The use of coupling mechanisms is a deceptively inefficient amplification strategy.

Flexures. The use of flexures at the ends of high load active elements is an inefficient method of obtaining rotational degrees of freedom.

Bending. Bending is a highly compliant method of carrying loads, such as in a lever and fulcrum design. Actuator designs where loads are transferred through components in bending should be avoided.

Compressive Pre-Load. To prolong active element lifetimes, it is imperative they stay in compression. Typically, this is done through a pre-load mechanism. Optimally, the pre-stress mechanism is placed in the actuation load path, performing the additional task of removing slop in load path joints and interfaces. Of course, compressive pre-stress values must be limited in order to avoid compressively depoling the active material.

Self-Reacting Actuators. By replacing the inert frame material that is used to react active material forces with other active material, larger mass efficiencies than those of stack/inert frame type actuators are theoretically possible. Such a design was pursued but later abandoned for a number of reasons, such as the fact that large stress concentrations occur at the interface between reacting elements, leading to actuator compliance. Furthermore, to keep the active material under compression, a complex pre-stress mechanism, outside the load path, is required for these actuators. Self reacting actuator designs, while theoretically efficient, do not offer much improvement over stack/inert frame designs and are complex to realize.

Simplicity. The actuator must be functional, meaning that the mechanism must be simple, for easy construction and to make it easier to track down problems. A complicated mechanism, while theoretically efficient, is often

practically impossible to realize.

Form Factor. The actuator should be compact or take up minimal space. In addition, it should actuate displacements in a desired direction.

Thermal Stability. The actuator should be thermally stable. Its performance should not vary greatly with temperature. Actuators need to operate over wide temperature ranges. A helicopter rotor blade, for example, has a thermal survival range that spans 185 deg F.

Linearity. Linear operation is desired so that the actuator can be modeled with standard linear techniques and easily incorporated into linear feedback control systems.

5 THE X-FRAME ACTUATOR DESIGN

The lessons learned from the actuator trade study led to the development of the X-Frame discrete actuator. An isometric drawing of the prototype is shown in Figure 6. This section presents the design issues associated with this prototype.

5.1 X-Frame Actuator operation

The actuator prototype consists of two piezoelectric stacks simultaneously extending against two steel frames. Each frame consists of two end plates connected by two side members. The frames are denoted as the inner and outer frames. The width of the inner frame is small enough to fit within the outer frame. At one end of the actuator, referred to as the pivot end, a roller pin is situated between the two frames. This pin allows relative rotation of the two frames while maintaining the distance between the frame end-plates. Stack extension and resultant frame rotation creates relative linear displacement between the frames at the opposite end of the actuator, referred to as the output end. As the stacks expand, the frames pivot on the roller pin and pinch together. By mounting the outer frame all of the displacement is realized at the output end of the inner frame. The small angle of the frame side members relative to the stack axes leads to a geometric stroke amplification. In the prototype an amplification factor of 15.2 was achieved.

A picture of the prototype in the lab is shown in Figure 7a. The output end of the outer frame was mounted to a 100 lb Interface load transducer and all the displacement was realized at the output end of the inner frame. A compressive pre-stress on the stacks is required to maintain the stack to frame contact. Such pre-stresses are easy to apply to the X-Frame actuator by placing a tensile load at the output. This pre-stress was accomplished in the prototype by hanging a 25 lb weight off the end of the actuator through a 0.024" diameter piano wire. This piano wire is shown in Figure 7a.

The operational characteristics of the actuator are easily understood through the use of a simple truss model, as shown in Figure 7b. Here the stacks are represented as vertical members of length l_s, and the frames as diagonal members of length l_f. The two frames are coupled by a horizontal member at the pivot side of the actuator, of length h_nom. The linearity of the device is based on the fact that the length of this horizontal member, and, therefore, the

angle q in Figure 7b, does not change appreciably during operation. The free amplification of the X-Frame actuator is found by solving, geometrically, for the displaced equilibrium of the truss given an induced stack extension. The exact free displacement is given as

(30)

Note that the derivation assumes that no stack end-caps are present. Equation (30) is an exact relationship and quantifies the linearity of the device, due to the weak dependence of the amplification on the induced strain.

The tip stiffness of the actuator, as with all the actuators considered in this study, was found using the principal of virtual work, also known as the dummy unit load method.²⁸ Using this method, the output stiffness of the truss in Figure 7b is

(31)

Note that only the compliance of the stacks and the adjacent frame side-members is included in this expression; the contribution of the horizontal member, h_nom was neglected. In practice, this member as well as various other components add compliance to the system. Using the parameters given below in Table 3, Equation (31) yields an expected actuator stiffness of 820 lb/in.

The d₃₃ constant for EC-98 material is 2.87 x 10^-5 mil/V (730 pm/V). Thus, a 57.4 V/mil electric field will induce a strain of 1650 microstrain. Large negative fields can cause heating and/or depole the EC-98 material.²¹ To achieve the desired electric field levels while avoiding large negative voltages, the maximum operational voltage consisted of a 400 V DC bias applied with a 600 V peak amplitude sine wave.

The X-Frame actuator design successfully integrates many of the optimal discrete actuator characteristics outlined in Section 4.4. First of all, the actuator design is uncomplicated. No flexures are used and the loads are predominantly transferred through axial loads in the frames and stacks. Because of this, the actuator retains a relatively high mass efficiency. Numerical optimization of this design, assuming perfect rolling contacts between stack and frame, yields a maximum achievable mass efficiency of 50%, matching that predicted by Equation (27) for steel frames and EC-98 stacks.

The form factor of the actuator is very good, such that a large stroke amplification is obtained with a relatively compact structure. In fact, the entire package is naturally suited for placement inside a rectangular cavity, such as

a rotor blade spar. This compact design also allows for large amplifications without substantial risk of buckling the stacks. Furthermore, because the amplified motion occurs transverse to the stack axes, the actuator is also ideally suited to the rotor blade application. The stacks are aligned and kept in compression under the centrifugal field in the rotorblades while the output motion occurs naturally in the chordwise direction.

5.2 Actuator manufacture

The piezoelectric stacks were fabricated by the EDO Corporation. The exact piezoelectric material used in the prototype is designated as EC-98, a PMN-PT ceramic. The stacks are composed of 140 layers, each 0.0221" thick, yielding an active material length of 3.094". Each layer is composed of a piezoelectric ceramic wafer, an electrode and two bond layers. The approximate thickness of the piezoelectric wafer in each layer is 0.0208". The stacks were fabricated with steel caps affixed to either end. Flat end-caps are used at the output end of the stacks because little relative rotation occurs between the stack and frame at this end, as discussed above. At the pivot end, where relative rotation does occur between the stacks and frames, cylindrical end-caps were used. These rolling contacts are used in lieu of flexures. The cylindrical and flat endcaps are approximately 0.1575" and 0.0225" thick, respectively, yielding an overall stack length of 3.274 inches.

It is important to machine each frame out of one piece of metal, to eliminate the additional compliance associated with using fasteners to connect the frame side members and end-plates. The prototype frames were milled out of stainless steel. The sharp corners inside the frames were made using a broaching process. An alternate method of fabrication is wire electron discharge machining (EDM), but this process was considered too costly for this initial prototype study.

Frames can also be made out of composites to boost the actuator mass efficiency, as discussed in Section 4.3.1. In fact, it may be possible to create temperature insensitive actuators by constructing frames out of metal matrix composites with a coefficient of thermal expansion (CTE) matching that of the active material. Preliminary calculations indicate that a frame composed of the metal matrix composite SiC/Ti can yield such a temperature insensitive design, while increasing the theoretical mass efficiency by 36%.

Table 3 gives the geometric properties of the prototype. D_s is the stack diameter.

Frames		Stacks
Material	Steel	Manufacturers	EDO Corporation
Modulus, Ef	29 x 10^6 psi	Material	EC-98
lf	3.304 in	Modulus, Ee	4.81 x 10^6 psi
Af	0.0377in^2	ls	3.094 in
hnom	0.446 in	Ds	0.315 in
Amp. Angle, q	7.766°	Capacitance	~ 350 nF

Table 3. Structural properties of the X-Frame actuator prototype

5.3 Scaling

The actuator is easily scaled for different applications. The amplification of the actuator is solely dependent on the angle the frames make with the stacks at the output end of the actuator. Upon scaling, as long as this angle remains constant, i.e., as long as all dimensions scale proportionally (geometric scaling), the amplification will remain constant. For example, micro-machining techniques could be developed to construct a miniature X-Frame actuator. Or, conversely, the actuator can be scaled for larger applications, e.g., incorporating the actuator into a full-scale CH-47D rotorblade for servo-flap control.

6 OPERATION AND PERFORMANCE OF THE X-FRAME ACTUATOR

This section presents the experimental performance of the X-Frame Actuator prototype. Because the performance of a discrete actuator is related to the load it drives, the presentation of the experimental results is prefaced with an impedance matching discussion.

6.1 Impedance matching

A simple linear model for a discrete actuator is

Q_a = K_a(q_a - q_f ) (32)

where Q_a is the output force, K_a is the output stiffness of the actuator, and q_a is the actuator displacement. Most active materials have important nonlinear effects that, strictly speaking, make the above model invalid. However, in many cases the materials are nearly linear. In any event, the model above provides a useful framework for determining the capability of a discrete actuator. Alternatively, Equation (32) may be written as

Q_a = K_aq_a + Q_ab (33)

where Q_ab = - K_aq_f is the blocked force capability of the actuator, i.e., the force produced by the actuator when the actuator motion is constrained to be zero. Q_ab may be thought of as an actuated force on the actuator. Now, suppose that the actuator displacement is constrained to be zero and the maximum field is applied to the actuator or, equivalently, that the maximum allowable actuation force is commanded. The actuator then behaves as a spring with spring constant K_a compressed by an amount q_f. Therefore, the actuator has internal mechanical energy, when fully actuated, of

(34)

In principle, all this energy can go into the actuation of a load. In practice, only a fraction of the energy can be converted into useful work. In particular, consider an elastic load with characteristic

Q_L = K_Lq_L (35)

where Q_L is the force applied to the load, q_L is the load displacement, and K_L is the load stiffness. If the load is connected directly to the actuator, the actuator and load displacements are equal (compatibility), and the actuator and load forces are equal and opposite (equilibrium), so that

q_L = q_a (36)

Q_L = - Q_a (37)

Substituting in the operating characteristics, Equation (32) and Equation (35), of the actuator and load yields

K_Lq_L = - K_a(q_L - q_f ) (38)

solving for the load deflection yields

(39)

The work done on the load is

(40)

(41)

Of course, the work done on the load is at most equal to the mechanical energy of the stack. Indeed, the maximum of W_L may be easily found, by differentiating Equation (41) with respect to K_L and setting the result to zero. This yields

K_L = K_a (42)

This is known as the impedance matching condition. ^5,7,12 For a load impedance matched to the actuator,

(43)

and

(44)

It is theoretically possible to transfer more of the strain energy of the actuator to the load using a mechanism with nonlinear gearing. In practice, however, such a mechanism would be exceedingly difficult to construct and would be undesirable for a number of reasons.

In some cases it may be desirable to operate with the actuator not impedance matched to the load. By using a very stiff actuator, the actuator will not deflect in response to varying load forces. For example, when controlling a helicopter servo flap, changing airloads on the flap may change the flap position, unless the actuator has significantly greater stiffness than the equivalent stiffness produced by the aerodynamics of the flap. Conversely, using a very compliant actuator will effectively command load force, rather than load deflection, which may be useful in some applications. However, changing the actuator stiffness away from the impedance-matched condition will always result in less energy transfer from the actuator to the load. The performance of the X-Frame actuator in driving an impedance matched load is considered in later sections.

6.2 Quasi-static performance of the actuator prototype

A discrete actuator is usually designed for use at frequencies below its first mode. Because of this, the actuator's quasi-static operation gives a good measure of its performance. Of course, at very low frequencies, £ 0:1 Hz, poling effects exaggerate the achievable strain in the active material. Thus, to capture the quasi-static behavior while minimizing poling effects, all data presented in this section was taken at 1 Hz.

The experimental performance of the actuator was determined by measuring actuator deformations while driving elastic loads of varying stiffness. This section presents the results from these tests.

6.2.1 Sinusoidal operation and hysteretic behavior

Figure 8 presents typical sinusoidal time histories of actuator deflection as a function of electric field for free actuation and while driving an elastic load of 390 lb/in. For each boundary condition the deflection characteristic is shown for two separate peak-to-peak electric field amplitudes. The low voltage case corresponds to a 15 V/mil DC bias applied in conjunction with a 15 V/mil peak amplitude sine wave. The high voltage case corresponds to a 20 V/mil DC bias applied in conjunction with a 30 V/mil peak amplitude sine wave.

Two important operational characteristics are highlighted from this data. The first noticeable characteristic is the non-trivial level of hysteresis. From the data the hysteresis appears to be directly related to the deflection of the material. Thus, the hysteresis is much smaller when driving an elastic load, such as in the servo-flap control application. Hysteresis is undesirable because it can cause substantial heating in the material, adds phase lag to the dynamic characteristics, and can lead to problems in static applications, such as blade tracking for helicopter rotor systems. The latter two problems can be overcome by closing simple feedback loops around the active materials; but the heating due to this phenomenon would still be present.²⁹

The second trend to notice from the data in Figure 8 is that the maximum deflection more than doubles for both the free and loaded case upon doubling the applied peak-to-peak electric field. This behavior is related to the nonlinear strain behavior of piezoelectric ceramics. This is discussed in more detail in Section 6.2.2.

6.2.2 Nonlinear strain behavior

The induced strain of a piezoelectric ceramic is related to the electric field, E, through the piezoelectric strain

parameter, d₃₃, as

e = Ed₃₃, (45)

At high applied fields most active materials exhibit a nonlinear strain characteristic such that these piezoelectric strain “constants" are, in fact, not constant. An excellent discussion of this effect is given by Crawley and Anderson.²⁹

Crawley and Anderson showed that the d₃₁ parameter for piezoelectric ceramics follows a strain dependent nonlinearity.²⁹ However, Fripp presents data for PMN-PT material, exhibiting a dominant electrostrictive effect, that shows electric field dependent nonlinearities.¹⁷

EC-98 is a PMN-PT type material, as discussed in Section 3 and the data in Figure 8 suggests a nonlinear strain behavior exists for this material as well. To determine the nonlinear characteristics, strain dependent and electric field dependent models were applied separately to predict actuator deflections. In each model the d₃₃ parameter was fit to the experimental free deflection (i.e., no load) data as a function of deflection and field, respectively. The deflection of the actuator driving a load was then predicted using these nonlinear d₃₃ models in conjunction with Equations (30), (39) and (45). Note that an iterative solution was used to find the appropriate d₃₃ in the strain dependent nonlinear model. This approach is very similar to that given by Crawley and Anderson.²⁹

Comparison of the actuator data to these two models while driving four different loads is shown in Figure 9a. As shown, the EC-98 material exhibits nonlinear behavior closer to the electric field dependent model. This indicates that, while the material does have a linear strain characteristic similar to a piezoelectric material, its nonlinear strain characteristic is closer to that of an electrostrictive ceramic.

The d₃₃ parameter as a function of applied field is backed out from the free deflection data by using Equations (30) and (45). It is shown in Figure 9b, along with the d₃₃ value reported by EDO.³⁰ As shown, this parameter varies by up to 50% from the reported value during operation.

6.2.3 Characteristic force/deflection load lines

Characteristic actuator load lines, such as those described by Equation (32), are found by measuring the actuator deflections while driving loads of varying elastic stiffness. How close this data approaches the linear model of Equation (32) gives a good measure of the actuator's linearity and stiffness properties. Varying stiffness loads were simulated by clamping the piano wire, shown in Figure 7a, at different locations along its length. The actuator was operated at each of these clamping conditions at 12 different applied field levels. The characteristic load lines from this data are shown in Figure 10. The abscissas represent the actuated displacement while the ordinates represent the actuated force. Each line corresponds to a different electric field level, according to the labels adjacent to the y-axis.

Over most of the operating range each force/stroke characteristic follows a linear trend, as shown by fitting the dashed line to the outer most actuator characteristic. This linearity is especially important at the center of the operating range (½ Q_ab, ½ q_f ), where an impedance matched load would operate having a force characteristic similar to the solid line noted in Figure 10 as the “Impedance Matched Load Characteristic". The intersection of the load characteristic with the actuator characteristic determines the actuator operating point. Notice that the two intersect within the substantially linear range of the actuator.

Comparing the fit of the outermost force/deflection characteristic to the “linear fit" highlights two nonlinear operating regimes for the actuator near the free (zero force) and blocked (zero deflection) boundaries. Larger deflections than predicted are realized at the blocked boundary condition because the stacks are under their largest compressive force at that point. The compressive force hardens the bond layers in the stacks, stiffening the actuator leading to larger actuator deflections. The nonlinearity at the free boundary condition is a result of the field dependent nonlinear characteristic of piezoelectric ceramics, discussed in Section 6.2.2.

Because the operating point is located away from these two nonlinear regimes, they should not significantly affect the performance. However, it is important to realize this nonlinear effect (especially at the free stroke operating point) when trying to extrapolate performance of systems incorporating piezoelectric ceramics.

The increase in electric field between each line is approximately 4.6 V/mil. The even spacing between the lines corresponding to each electric field demonstrates that the stacks are not saturating, even at high applied fields. According to Equation (45), a 54.75 V/mil induces a strain of 1574 microstrain, using the d₃₃ value reported by EDO for EC-98.³⁰ The data indicates that the EC-98 stacks exhibit little saturation at this strain level, supporting the validity of the strain assumption made in Table 1.

6.2.4 Performance calculation

Taking the area under the straight line fit to the outermost actuator characteristic in Figure 10 gives a good measure of the actuator energy available for linear operation. This dashed line intersects the axes indicating a linear peak-to-peak blocked force of 35.8 lb and free peak-to-peak deflection of 81.0 mil, yielding an output energy of

W_a = 1.45 in-lb (46)

The mass of the entire actuator prototype 0.00830 slug (121 g), giving the actuator output energy density as

(47)

The mechanical and mass efficiencies of the device are found by normalizing this energy density by the active material strain energy and energy density given by

(48)

and Equation (3), respectively. One problem in evaluating these relations is that the d₃₃ parameter (and e) varies as a function of field, as shown in Figure 9b. The cumulative strain energy in the active material could be estimated by integrating Figure 9b over the appropriate boundary conditions, but this calculation would be an inferential measure of stack performance. For example, it may be that the force/deflection characteristics of the stacks alone could exhibit much larger nonlinear characteristics than those of the actuator shown in Figure 10. To make an accurate estimate of the available linear strain energy in the active material, force/deflection characteristics of the stacks, similar to those of Figure 10, must be obtained. A linear estimate could be fit to these characteristic lines as above and the available linear active material strain energy could be calculated. A component tester designed to acquire this data is currently under development in the Active Materials and Structures Lab (AMSL) at MIT. In the absence of such active material strain energy data, the catalog d₃₃ value for the PMN-PT stacks of 2.874 x 10^-5 mil/V is used,³⁰ and an applied field level of 54.75 V/mil is assumed for all calculations in this section.

In addition to determining e in Equation (48), the calculation of the active material volume, V_e, and active material mass also affect the efficiency calculations. The mass and mechanical efficiency calculations for the actuator are performed using two different approaches. These two calculations result in upper and lower bounds to the experimental actuator efficiencies. The first method assumes that the “active" material is just the piezoelectric ceramic. Thus, the additional mass from end-caps, electrodes, electrode bus and the additional compliance from the stack bond-layers are accounted for as actuator losses. This first method results in a lower, conservative, bound to the actuator efficiency. The second method takes the opposite approach, where all additional mass associated with the stacks is taken as “active material mass". Furthermore, the bond-layer losses are also taken into account as stack losses and not actuator losses. The second method is more realistic but may over-predict the actuator efficiency somewhat. This second method gives an idea of the achievable actuator mass efficiency if 100% mass efficient stacks were used in place of those supplied by EDO. The calculations are as follows.

The active material element volume is the volume of just the piezoelectric material. It is the same in both methods.

(49)

In method 1, the strain energy is assumed to be the strain energy in the bulk material, using Equation (48), it is

(50)

The active material mass for method 1 is just the mass of the piezoelectric material. Multiplying the density for EC-98 material, of 15.23 slug/ft³, by the active material volume, Equation (49), yields

(51)

The energy density of the material is found by dividing the strain energy by the mass, giving

(52)

Note that this energy density agrees with that reported in Table 1 for bulk EC-98 material when scaled by the ratio of strains, (1650/1574)².

In method 2 the bond layer losses are accounted for in the strain energy calculation, so

(53)

The active material mass used is the entire mass of both stacks, including electrode bus and endcaps. The mass of the two stacks were found by weighing them, yielding

(54)

Dividing Equation (53) by the mass, Equation (54), gives the active material energy density for this method

(55)

Again, this energy density also agrees with that given in Table 1 for EC-98 stacks when scaled by the induced strain.

The mass and mechanical efficiencies are found for each method by dividing the corresponding actuator output energy and energy density by the active material strain energy and energy density, respectively. The results are shown in Table 4, along with the associated optimal mass efficiency for each case according to Equation (27).

Method #	We In-lb	Me slug	Ue ft-lb slug	hmass	h*mass	hmech	h*mech
1	3.86	0.00400	80.5	18.1%	45.2%	37.5%	67.2%
2	2.70	0.00475	47.4	30.7%	50.5%	53.6%	71.0%

Table 4. Comparison of measured efficiencies calculated using two varying methods

Note that because the modulus of the stacks for the two methods is different (bond layer losses are accounted for in Method 2,

but not in Method 1), the optimal mass efficiency, Equation (27), also changes. The important fact is that because the losses are accounted differently in the two methods, the calculated efficiencies differ. But, the product of the mass efficiency and the energy density in both cases is equal because the energy density of the actuator is a constant, given by Equation (47). This range in mass efficiencies is given because it is impossible to discern the true energy output of the active material elements from these tests.

6.2.5 Actuator losses

As discussed in Section 4.3.1, losses in the stacks and the frame members adjacent to the stacks are expected. However, other compliance losses occur in practice. Furthermore, as a consequence of the mass efficiency definition,

even the presence of inert frame material such as the cylindrical end caps and frame end-plates lowers the mass efficiency. The following list gives the estimated sources of the additional loss.

Eccentric Loading. Eccentric loading of the stacks introduces bending stresses in the material. As discussed previously, bending stresses are a very compliant way to carry loads. These eccentric losses can severely affect actuator performance. The data presented above was obtained after careful alignment of the stacks within the frames.

Experiments showed that the flat/cylindrical endcap combination in the stack prototypes exacerbated the eccentric loading condition.

Bond Layers. The bond layers in the piezoelectric stacks reduce the effective stack stiffnesses by at least 70%. This is an active material issue and not at all linked to the X-Frame actuator design. Co-fired stacks may offer lower compliance losses but there still exist lifetime questions regarding these stack designs.

Hertzian Losses. Some Hertzian losses occur at the interface between the stacks and frame end-plates. In the prototype, these losses result in additional compliances (< 10%). These losses are small in comparison to those that would exist if flexures were used to create these rotational degrees of freedom.

Frame Spanning Losses. Losses occur at the frame end plates due to bending. These losses are unavoidable because the frames must straddle the stacks. These bending losses are estimated to be about 11%.

6.2.6 Impedance matching data

An alternate method of viewing the data shown in Figure 10 is by examining the energy transferred from the actuator into the load. Such a plot is shown in Figure 11.
For each electric field level, the energy delivered to the load, Equation (40), is plotted for each clamp position. For each electric field level the expected impedance matching curve, given by Equation (41), is fit to the data by adjusting K_a and q_f in a least squares fashion.

As expected and as discussed above in Section 6.1, the work transferred to the load is a minimum when the stiffness of the load is much higher or much lower than the actuator stiffness. The optimum transfer of actuator energy to the load occurs at the impedance matching point, where K_a = K_L.

This procedure gives a least squares actuator stiffness of 467 lb/in. However, direct measurement of the short circuit actuator deflection given an applied external load yields a stiffness of 590 lb/in. The difference between these two measurements may indicate that EC-98 exhibits a field dependent Young's modulus. Further research is needed to identify the cause for this difference.

6.3 Dynamic actuator characteristic

The bandwidth of the actuator defines the frequencies over which control can occur. A transfer function of the prototype is shown in Figure 12.
The transfer function was taken with the piano wire clamped such that the actuator drove a nearly impedance matched load at its output. The exact spring load driven was 777 lb/in. Even with the wire clamped, a number of piano modes were present during the transfer function test and are evident in Figure 12 by the nearly unobservable modes at approximately 100, 220, and 370 Hz. The magnitude of the transfer function is normalized to give output stroke [mils] per unit applied electric field [V/mil].

This transfer function shows the first mode of the actuator at about 543 Hz. The goal is to scale this actuator by a factor of 2/3 for incorporation into a model scale helicopter rotorblade. Thus, the first mode in an equivalent model scale actuator will increase by 1.5. Of course, once the actuator is connected to a trailing edge servo-flap, the inertial characteristics of the flap will lower the first mode frequency. However, a model scale actuator/servo-flap system should yield a bandwidth between 100-200 Hz, which meets the requirements stated in Section 2.

6.4 Estimate of actuator mass

The work done in driving a trailing edge flap located at the tip of a rotor blade is

(56)

where q is the dynamic pressure at the tip, c is the blade chord, is the servo-flap hinge moment curve slope and is the peak-to-peak flap deflection magnitude. For a 15% of chord flap, XFOIL simulations give = 3.4 x 10^-4 deg^-1. ³¹

The full scale actuator mass required to perform this amount of work is estimated using an impedance matching efficiency of 0.25 and the experimentally determined actuator energy density, given in Equation (47), as

(57)

Additional mass will be necessary to couple the actuator to the blade. Preliminary designs show that this extra mass can increase the total actuator mass by about 70%, resulting in a spanwise mass of 18.8 lbm/ft.

The mass of an operating Chinook blade is 10 lbm/ft. Assuming the actuator powers a flap of equal length, a 10% of span flap will increase the blade weight by 19%, which is within the requirements set in Section 2.

7 CONCLUSIONS

In this paper, we have considered the important issues in the design of actuators based on active materials, in particular for rotorcraft applications. Especially in aerospace applications where weight is an important consideration, the energy density of the actuator is a critical metric in evaluating potential actuator designs. High energy density actuators require the use of high energy density materials. Given the state-of-the-art, the preferred material is piezoelectric ceramic, using the direct piezoelectric effect (i.e., 33-actuation). This is most easily accomplished using piezoelectric ceramic stacks.

Furthermore, high energy density actuation requires high mass-efficiency amplification mechanisms. Based on our survey of existing and proposed amplification devices, we concluded that a number of approaches should be avoided, including planar (bender) actuators, coupling mechanisms, and self-reacting actuators, which generally have high elastic losses, even when well-designed. Generally, the approach likely to achieve the highest energy density is an actuator where piezoelectric stacks react against an inert frame, with no flexures in the load path, and a simple amplification mechanism.

Based on this understanding, we introduced a new device, the X-Frame Actuator. Theoretical and experimental results show that this device has a mass efficiency close to the theoretical maximum for stack/inert frame devices, and significantly better than both commercially available and newly developed actuators reported in the literature. Although self-reacting actuators can in theory achieve 100% mass efficiency, practical considerations, such as the need to apply a pre-load, generally limit the mass efficiency of these devices to less than that of stack/inert frame devices.

Finally, in addition to possessing a high energy density, the actuator also satisfies a number of other requirements. For instance, it has a compact form factor and very linear operational characteristics. These combined benefits make the X-Frame Actuator ideal for most applications requiring fast acting, large stroke actuation.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the following people for their contributions to this research: (At MIT) Professor Nesbitt Hagood, Paul Bauer, Dr. John Rodgers, Dr. Aaron Bent, Michael Fripp, Dr. Kamyar Ghandi, Professor Mark Spearing, Winston Fan, Corinne Ilvedson, and Ben Erwin. (At Boeing) Robert Derham, Leo Dadone, Doug Weems, and Dean Jacot. This research was supported by DARPA under Air Force Contract No F49620-95-2-0097 with Dr. Spencer Wu of AFOSR and Dr. Robert Crowe of DARPA serving as technical monitors.

REFERENCES

1. S. R. Hall, K. Y. Yang, and K. C. Hall, “Helicopter rotor lift distributions for minimum induced power loss," Journal of Aircraft 31, pp. 837-845, July-August 1994.

2. J. C. Garcia, “Active helicopter rotor control using blade-mounted actuators," Master's thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, Feb. 1994.

3. T. A. Millot and P. P. Friedmann, “Vibration reduction in helicopter rotors using an actively controlled partial span trailing edge flap located on the blade," Tech. Rep. 4611, NASA, June 1994.

4. R. L. Spangler, “Piezoelectric actuators for helicopter rotor control," Master's thesis, Massachusetts Institute of Technology, Cambridge, MA, June 1989.

5. R. L. Spangler and S. R. Hall, “Piezoelectric actuators for helicopter rotor control," in 31st Structures, Structural Dynamics and Materials Conference, (Long Beach, CA), Apr. 1990.

6. S. R. Hall and R. L. Spangler, “Piezoelectric helicopter blade flap actuator," July 1993. U.S. Patent No. 5,244,826.

7. S. R. Hall and E. F. Prechtl, “Development of a piezoelectric servo flap for helicopter rotor control,"Journal of Smart Materials and Structures 5, pp. 26-34, Feb. 1996.

8. C. Walz and I. Chopra, “Design and testing of a helicopter rotor model with smart trailing edge flaps," in 35^th Structures, Structural Dynamics and Materials Conference, Apr. 1994.

9. O. Ben-Zeev and I. Chopra, “Development of an improved helicopter rotor model with smart trailing-edge flaps for vibration suppression," in SPIE Smart Structures and Integrated Systems, Feb-Mar 1995.

10. C. M. Bothwell, R. Chandra, and I. Chopra, “Torsional actuation with extension-torsion composite coupling and magnetostrictive actuators," 33, Apr. 1995.

11. Physik Instrumente, Auburn, MA, Products for Micropositioning, Catalog US-Edition, 1995.

12. V. Giurgiutiu, Z. Chaudhry, and C. A. Rogers, “Issues in the design and experimentation of induced-strain actuators for rotor blade aeroelastic control," in Tenth VPI & SU Symposium on Structural Dynamics and Control, (Blacksburg, VA), May 8-10 1995.

13. P. Jänker, F. Hermle, T. Lorkowski, S. Storm, and M. Wettemann, “Development of high performance piezoelectric actuators for transport systems," in 6th International Conference on New Actuators, June 1998.

14. F. K. Straub and A. A. Hassan, “Aeromechanic consideration in the design of a rotor with smart material actuated trailing edge flaps," in 52nd Annual Forum of the American Helicopter Society, (Washington D.C.), June 1996.

15. V. Giurgiutiu, Z. Chaudhry, and C. A. Rogers, “Energy-based comparison of solid-state actuators," Tech. Rep. CIMSS 95-101, Center for Intelligent Material Systems and Structures, Sept. 1995.

16. A. A. Bent and N. W. Hagood, “Piezoelectric fiber composites with interdigitated electrodes," submitted to the Journal of Intelligent Material Systems & Structures , Mar. 1996.

17. M. Fripp, “Distributed structural actuation and control with electrostrictors," Master's thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, Jan. 1995.

18. A. A. Bent, Active Fiber Composites for Structural Actuation. PhD thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, Jan. 1997.

19. K. H. Chan, “Nonlinear modeling of high field ferroelectric ceramics for structural actuation," Master's thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, July 1994.

20. ETREMA Products Inc., Ames, IA, ETREMA Terfenol-D Magnetostrictive Actuators.

21. G. Cook, 1995-1996. EDO Corporation, Personal correspondence.

22. M. Lutz, “Piezoelectric actuator control of a helicopter rotor." MIT Department of Aeronautics and Astronautics, 16.622 Project Report, Course Advisor: Professor Steven Hall, Fall, 1995.

23. C. Ilvedson, “Piezoelectric actuator control of a helicopter rotor." MIT Department of Aeronautics and Astronautics, 16.622 Project Report, Course Advisor: Professor Steven Hall, Fall, 1995.

24. R. L. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, Dover Publications, Inc., New York, 1975.

25. M. F. Ashby, Materials Selection in Mechanical Design, Pergamon Press, New York, 1993.

26. R. C. Fenn, J. R. Downer, D. A. Bushko, V. Gondhalekar, and N. D. Ham, “Terfenol-d driven flaps for helicopter vibration reduction," in SPIE Smart Structures and Intelligent Systems, vol. 1917, pp. 407-418, 1993.

27. P. H. Stahlhuth, “Piezoelectric stack motor stroke amplifier," September 1988. U.S. Patent No. 4,769,569.

28. R. L. Bisplinghoff, J. W. Mar, and T. H. H. Pian, Statics of Deformable Solids, Dover Publications, Inc., New York, 1990.

29. E. F. Crawley and E. H. Anderson, “Detailed models of piezoceramic actuation of beams," The Journal of Intelligent Material Systems and Structures 1(1), pp. 4-25, 1990.

30. EDO Corporation, Acoustics Division, Ceramic Operations, 2645 South 300 West, Salt Lake City, UT, Piezoelectric Ceramics, Material Specifications, Typical Applications.

31. M. Drela, “Xfoil: An analysis and design system for low reynolds number airfoils," in Springer-Verlag Lecture Notes in Engineering - Low Reynolds Number Aerodynamics, 1989.