Bichop R.H. (Ed.) Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling

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21-52 Mechatronic Systems, Sensors, and Actuators

where S is strain, T is stress (N/m

), E is electric field strength (V/m), D is electric displacement (C/m

is the compliance of the material (m

/N) when the electric field strength is constant and

is the

permittivity (F/m) under constant stress, and d is piezoelectric constant (m/V or C/N). Superscript t

stands for matrix transpose.

The first members on the right side of the equations refer to the mechanical properties of an elastic

body (Equation 21.31) and to the electric properties of a medium (Equation 21.32), while the second

ones describe the piezoeffect.

Piezoelectric properties of piezoelectric materials depend on the directions of electrical and mechanical

inputs/outputs. Thus, these properties are described by constants with two subscripts, first of which is

related to electrical and second to mechanical action or response directions in the orthogonal axes X, Y,

and Z system. Indexes 1, 2, and 3 are prescribed to these axes respectively. Piezoelectric material polar-

ization direction coincides with that of the poling electric field and is referred by convention to the axis

Z with index 3. Subscripts 4, 5, and 6 are used additionally for describing shear distortions in respect to

the directions X, Y, and Z (Figure 21.81).

Indexes show possible piezomaterial operation mode thickness expansion, transverse expansion, thick-

ness shear, and face shear. The mode of motion depends on the location of electrodes relative to the

polarization axis and the displacement that is used. The poling electric field direction causes elongation

in this direction and contraction in the perpendicular ones, which are three times smaller than the first

one. The reverse field causes contraction along the electric field direction; the elongation in perpendicular

directions and displacement proportions are the same. The piezomaterial properties are identical along

axes 1 and 2. The main constants characterizing the piezoeffect are:

•

(piezoelectric constants) strain or charge coefficients expressed in C/N or m/V (according to

direct or reverse piezoeffect). They relate to the strain developed by the electric field in the absence

of mechanical stress (Equation 21.31), and to the electric charge per unit area by the applied stress

under zero electric field (Equation 21.32). Example: symbol d31 means that electrodes are per-

pendicular to axis 3 (electric field along it) and stress or strain is along axis 1.

•

(voltage constants) relate to open circuit electric field developed per applied mechanical stress

or strain developed per applied charge density and is expressed in Vm/N or m

/C. The relation

between d

and g

is as follows:

(21.33)

•

(electromechanical coupling factors)—energy ratios describing conversion from mechanical to

electrical energy and vice versa. Factor k

is the ratio of converted energy to input energy at

operating frequencies far from resonant.

(21.34)

•

Factor k

refers to the plane mode operation (strain or stress is equal in all directions perpendicular

to axis 3).

FIGURE 21.81 Directions for indexing piezoelec-

tric constants.

X (1)

Z (3)

Y (2)

Polarization

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Actuators 21-53

21.3.3 Piezomaterials

Materials that exhibit a significant and useful piezoelectric effect fall into three main groups: natural

(quartz, tourmaline, topaz, Rochelle salt) and synthetic crystals (lithium sulfate, ammonium dihydrogen

phosphate), polarized ferroelectric ceramics, and certain polymer films. The main piezomaterials for

engineering applications are ferroelectric ceramics and PZT (lead zirconate titanate). The latter is char-

acterized by high coupling factors, and piezoelectric and dielectric constants over extended temperature

and stress ranges. Barium titanate, lead magnesium niobate, modified lead titanate, and bismuth titanate

compounds are used, as well as other special compositions. Because of their natural asymmetric structure,

crystal materials exhibit the effect without further processing. After sintering ferroelectric ceramics,

domains are of statistically distributed orientation and must be artificially polarized by a strong electric

field, while the material is heated above its Curie point (the temperature at which a piezomaterial becomes

completely depolarized) and then slowly cooled with the field applied. The remnant polarization being

retained, the material exhibits the piezoeffect. Piezopolymer—polyvinylidene fluoride (PVDF) is a special

class of polymer that exhibits a high degree of piezoelectric activity. They are used for manufacturing

piezofilms of low thickness (less than 30 mm), which may be laminated on the structural materials. The

values of piezo constants for PVDF piezofilms are lower than those for piezoceramics.

New piezoelectric materials such as single crystals and relaxors are now under research. Single crystals

of natural or man-made materials exhibit the piezoelectric properties that might be offered by polycrys-

talline element (ferroelectric ceramics) with perfectly aligned domains, which is impossible; thus, single

crystals demonstrate improved performance. For example, single crystal lead magnesium niobate lead

titanate exhibits ten times the strain of comparable polycrystalline PZT element. A large variety of

materials, such as lithium niobate, lead zirconate, niobate lead titanate, lithium tetraborate, and so forth,

are used to fabricate single crystal piezoelectric elements. Relaxor materials such as lead nickel niobate

are less sensitive to temperature than the conventional ones, because the transition between piezoelectric

behavior and the loss of such a capability cover some temperature range (Curie range) and do not occur

at a specific Curie point. They exhibit higher electromechanical coupling factors than conventional

ceramics (greater than 0.9), which makes them attractive for actuators, transducers, and so forth.

Polypropylene foils with foam structure, caused by the enclosed micrometer scale vapor locks, exhibit

piezoelectric behavior after polarization, and are being introduced owing to higher piezoelectric constants

values than polyvinylidene fluoride.

21.3.4 Piezoactuating Elements

Piezoactuating elements are produced in the wide range of sizes as squares, rectangles, rings, disks,

spheres, hemispheres, bars, cylinders, and special elements. The typical thickness is from 0.2–20 mm and

up to 100 mm length or external diameter. Due to the unique properties of the material, they can be

used directly as actuators. The main disadvantages are the dependency of characteristic values on the age

and temperature and the necessity of high- voltage power supply.

The piezoeffect is linearly dependent on the applied electric field. Possible strength depends on

piezomaterial short circuit resistance and is in the range 1–2 kV/mm. Maximum possible relative change

in length is of the order of 0.1%. The shortest time of expansion to reach the nominal displacement is

1/3 of the period at resonant frequency oscillations of the mechanical system containing a piezoelement.

The piezoelement resonant properties are described by frequency constant Ni, which is the resonance

frequency fr multiplied by the linear dimension of the piezoelement related to the specific resonance.

Voltage being changed (it remains constant after the change), the piezomaterial continues to

expand/contract in the same direction. This drift, known as creep, can be estimated by

(21.35)

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21-54 Mechatronic Systems, Sensors, and Actuators

where ∆L is 0.1 s expansion after the positioning process and

is the drift factor. It depends on the design

and mechanical load and lies between 0.01 and 0.02. Hysteresis is common in the piezomaterials as well.

PZT hysteresis is a fairly constant fraction of the stroke, and the width of the hysteresis curve for it can

be as large as 20% of the stroke. Due to compensation strategies, hysteresis errors decrease down to 3%.

A piezoelement with electrodes laminated onto it is an electrical capacitor. Because of extremely high

piezomaterial internal resistance (more than 100 MOhm), only small discharge current flows if piezo-

material remains static in the expanded state. Thus, the piezoelement is separated from the source of

high voltage, and its expansion decreases slowly. This in turn causes a change in the charge, which results

in a current.

(21.36)

where Q is the charge, C is the capacity, and V is the voltage.

The same equation is valid to describe the flow of electric energy when the material is experiencing a

change in strain, and to define the time required to build up voltage in the piezoelement.

Table 21.4 shows different piezoelement sensing/actuating possibilities for some shape cases. Note: F

is the force and is dielectric permittivity of the material at constant stress in direction 3, K

is relative

dielectric constant (K

), and

is dielectric permittivity in vacuum.

Piezoactuators are able to generate a force of several thousand Newtons in a stroke range of hundreds

micrometers (without amplifiers) with sub-nanometer resolution and response time of millisecond

range. Resonant frequencies of industrially applied actuators are in the range from several to hundreds

kiloHertz. The basic types of piezoactuators used are stacked and are of laminar design. Stacked actuators

consist of some thin layers of piezoactive material between metallic electrodes in parallel connection

(Figure 21.82a).

This way of connection allows greater stroke at lower voltage. Stacked actuators may be produced by

cut-and-bond and tape-casting methods. The first one is realized by stacking a large number of thin

piezoelectric disks with copper shims in between each disk and positive electrodes to opposite sides of

the actuator stack. Usually, these disks are 0.3–1 mm thick. Displacement in this case is in proportion

to the number n of disks, if no external load is applied and operating mode is d

(21.37)

Relationship between the piezoactuator’s generated displacement and external load

is a linear

func

tion

(21.38)

where F

and c are blocked force and piezoactuator stiffness.

Blocked force is the force that is generated at maximum voltage when the actuator is not allowed to

move. The displacement attained at maximum voltage when the actuator is not loaded is called free

displacement.

If the piezoactuator is loaded by spring, the following relationship is valid:

(21.39)

where ∆L

max

is free displacement and c

is the spring stiffness.

Such stacked actuators allow us to achieve the stroke of 100-µm range. The principal disadvantage of

such technology is high voltage, necessary to get sufficient electric field strength, because the minimum

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Actuators 21-55

TABLE 21.4 Piezoelement Sensing/Actuators Possibilities

Action Mode (L, length; W, width;

T, thickness; D, diameter) Generated Voltage, V

Displacement, ∆L

(∆T,∆D,∆X) Capacitance, C

Note: F is the force and is dielectric permittivity of the material at constant stress in direction 3, is

relative dielectric constant ( ), and

is dielectric permittivity in vacuum.

Transverse length mode: L> 3W>3T

∆

Thickness extension mode: D >5T

∆

VdT

Radial mode: D >5T

Not applied

Longitudinal mode: L >3D

Vdl

Thickness shear mode: W>5T, L>5T

Vdx

Radial mode of a thin

wall cylinder:

>8T

pDgV

031

)/ln(

Note: D

, D

and D

are

outer, inner and mean

diameters of the cylinder;

p is the pressure.

Longitudinal mode of a

thin wall cylinder:

L>5T

= /

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21-56 Mechatronic Systems, Sensors, and Actuators

layer thickness is limited. The high-voltage actuators, consisting of 0.5 mm thickness discs disks, require

about 1000 V and the low-voltage ones, consisting of 0.1-mm thickness disks, require about 200 V electric

field.

Tape-casting technologies make it possible to harness the full potential of the piezoelectric effect at

low voltages, enabling to reach the maximum permissible field strength already at 100 V and less. This

technology uses a number of tape-cast ceramic layers with thickness of several tens of micrometers

laminated together with screen-printed internal electrodes for producing multi-layer piezoactuating

elements. External connections are provided to the internal electrodes by screen printing conductive

materials onto the sides of the ceramic laminate. It allows to get much thinner ceramic layers, a customized

internal structure, and flexibility of the component design. All this opens new possibilities for high volume

and low-cost production. Piezolectric multilayer elements can be manufactured from a number of

different materials, each optimized to match application-specific functions and performance. Ceramic

multilayer actuators may be made very small (e.g., 1 mm × 1 mm × 0.2 mm), give up to 5

µm stroke,

and control up to 20 kN forces with resonant frequency 0.5–10 MHz. The produced multi-layer ceramic

may be used as a finished transducer and in stacked design.

Laminar design actuators are based on transversal effect (mode d

) and consist of piezoelectric layers

with films bonded on to the structure (Figure 21.82b). They are of lower stiffness and displacement than

the stacked ones. The actuator displacement and stiffness depend on the length/thickness ratio, and its

influence is opposite. Therefore piles consisting of several strips are often used in making a design similar

to the stacked design.

If a long stroke is required, bending piezoelements based on transversal effect can be used. All kind

of bending elements feature greater deflection, lower stiffness, lower resonant frequency, response speed,

and life time in comparison with stacked design. A unimorph and a bimorph are devices of such kind,

which allow the displacement to reach up to 1000 µm. A unimorph is a composite cantilever of two

layers (Figure 21.82c). One of them, an elastic shim, is of structural material and the other is of piezo-

material. The shim is of constant length and the element bends in order to compensate the different

behavior when the ceramics expand or contract. Two ceramic strips with the shim in between are used

FIGURE 21.82 Piezoactuators: (a) stacked; (b) laminated; (c) unimorph; (d) amplified; (e) mooney; and

(f) inchworm motor.

(a) (b) (c)

(d) (e) (f)

PZT

Driven link

23 1

∆

PZT

Steel (PZT)

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Actuators 21-57

in bimorph, and one of them expands while the second one contracts. Disk or ring multilayer-bending

piezoactuators consist of circular elements with a diameter of a few centimeters. When voltage is applied,

the component deforms, taking partly a spherical shape, and reaches displacement up to 200 µm.

Displacement amplification up to 2000 µm can be achieved by various constructive means. The stiffness

of such designs decreases with the square of the displacement amplification ratio and is much smaller

than in the stack design. Along with levers (Figure 21.82d), monolithic hinge lever mechanisms made

of structural material by cutting constricted hinges are used to function as lever mechanisms. The second

mechanisms are free of backlash. Moonie piezoactuator consists of a piezoelectric disk sandwiched

between two metal plates (Figure 21.82e) with cavities bonded together. The small radial displacement

of the disk is transformed into a much longer axial displacement, normal to the surface of the metal

plates. Forces and displacements achieved are in between conventional multi-layer and bimorph actuators.

If absolute positioning of high accuracy is necessary, the controlled operation of piezoactuators is used.

Sensors detect the actual displacement, and the input voltage is controlled by their signals. This leads to

positioning with high accuracy, avoiding hysteresis effect and changes in position because of fluctuating

forces.

An inchworm motor is used for positioning in hundreds of mm range with resolution up to 2 nm

and up to 2 mm/s speed. This motor is composed of three piezoceramics elements (Figure 21.82f). Two

elements (1 and 2) are used to clamp and release the driven link, and element 3 is fixed on the frame

and used to shift the driven link when expanding. By applying drive voltages with a certain phase

difference, this device can move simulating the worm motion, and the working cycle of stepping is as

follows: 1e; 3e/ driven link moves by step; 2e; 1c; 3c/ driven link moves by step; 2c. Here e stands for

expansion and c for contraction.

21.3.5 Application Areas

Owing to the inherent properties of piezomaterials, actuators with a lot of engineering advantages can

be developed. Some of them are: compactness and light weight, rapid response, practically unlimited

resolution, no magnetic field, large force generation, broad operating frequency range, high stability,

solid state, low power consumption, proportional to the applied voltage displacement, and 50% and

more energy conversion efficiency.

They are used in micromanipulation, noise and vibration suppression systems, valves, lasers and

optics, ultrasonic motors, positioning devices, relays, pumps, in automotive industry, industrial automa-

tion systems, telecommunications, computers, and so forth. Some of the applications are shown in

Figure 21.83.

Suppression of oscillations: Piezoactive material-based dampers convert mechanical oscillations into

electrical energy. The generated energy is then shunted to dissipate the energy as heat, that is, oscillation

energy is dispersed. The principle scheme is given in [2].

Microrobot: The microrobot platform legs are piezoactuators. By applying voltage to the electrodes,

the piezo–legs are lengthened, shortened, or bent in any direction in a fine movement.

Micropump: The diaphragm is actuated by a piezoactuator; input and output check valves are subse-

quently opened for liquid or gas pumping. The advantages are fast switching and high compression rate.

Microgripper: The piezoactuator works on contraction for gripping motion based on the compliant

mechanism. The gripper is of very small size and almost of any required geometrical shape.

Micromanipulator: Due to the practically unlimited resolution, piezoactuators are used in numerous

positioning applications.

Microdosage device: Piezoactuators allow high precision dosage of wide variety of liquids in the range

of nanoliters for various applications.

Multilayer ceramics and single crystals are expanding their application. Multi-layer ceramics as finished

transducers are used for stabilization of hard drives, tuning the lasers, adjusting mirrors in optical switches,

and other instrumentation applications. Their stacked design is applied for fuel injection, optical switches,

and medical instrumentation where nanorange is required. Single crystals have found applications in

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21-58 Mechatronic Systems, Sensors, and Actuators

acoustical, optical, and wireless communication as well as actuators. Some of them exhibit useful combi-

nations of electro-optic and piezoelectric properties and are exploited for surface acoustic wave and electro-

optic devices.

21.3.6 Piezomotors (Ultrasonic Motors)

Ultrasonic motors, from the first attempt to design them by Williams and Brown in 1942 [3] and first

developments in Eastern Europe in the early 1970’s [4–6], up to commercializing them in Japan in the

last decade of the XX twentieth century [7–9], find wider and wider applications as converters of high-

frequency mechanical oscillations (dozens of kHz) into continuous motion. If piezomaterials are used

as oscillators, such motors may be called piezomotors. Various types of piezomotors may be developed,

and the basic ideas are given in Table 21.5. Piezomotors are mainly based on producing elliptical motion

in the contact area between input and output links. The oscillations in the contact point of the oscillating

stator and the driven link are excited in them in such a way that the trajectory of this point is elliptical,

and thus the friction is used to involve this link into the motion. The shape of trajectory and frequency

predetermines the velocity of the driven link. For this purpose, oblique impact on the output link or the

traveling wave is made use of.

In piezomotors, making use of oblique impact, friction force transmits motion and energy between

the input and output links. This may be realized by two oscillatory motions (normal and tangential

components) u

and u

in the contact area with phase difference , which is used also to change the

direction of the output link motion. Both motions can be realized by one or two active links oscillating

resonantly. Various oscillations (longitudinal, transversal, shear, and torsional) offer possibilities to

develop different kinds of piezomotors.

Traveling wave motion piezomotors are based on frictional interaction between the traveling wave

motion in the elastic body and the output link, that is, its principle of operation is similar to the harmonic

traction transmission. Wave propagating along the surface (Rayleigh wave) of the input link forms the

elliptical motion in the contact area. Rayleigh wave is a coupled wave of longitudinal and shear waves,

thus each surface point in the elastic medium moves along an elliptical locus. Flexural, shear, torsional,

and longitudinal waves are used. Traveling wave in the piezoceramic transducer is excited by the electrical

(a)

FIGURE 21.83 Applications of piezoactuators.

(a)

Damped structure

PZT

(b) (c)

PZT

Channel to

reservoir

Nozzle

Dossage chamber

PZT

Platform

PZT

PZTValves

Input Output

Diaphragm

PZT

Platform

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Actuators 21-59

field. It can be generated by two standing waves whose phases differ by 90⬚ to each other, both in time

and in space:

(21.40)

In practice, this is accomplished by dividing the electrodes of the converters into n equal parts and

connecting them to the n-phase generator of electrical oscillations, where n $ 3; phases are shifted

between adjacent electrodes being 2π/n, or by using discrete converters. Application of piezoceramics

sectors with alternating polarization allows us to simplify design. Generation of more than two standing

waves increases output power.

Piezomotors can be developed by using only one standing wave, excited in the stator that moves

its vibratory tips, generating flat-elliptical movement when they bend because of the restriction by

the rotor. A one-phase driving voltage signal is needed and electric circuits are less complicated than in

the traveling wave case, but alternating the direction of the output link rotation can sometimes be

complicated.

Piezomotors with frictional anisotropy of contact are based on oscillatory motion variations in the

normal direction of active link contact during the oscillation cycle. This is achieved by superposing

additional periodic actions in the contact. The distinguishing feature is

␶

/T, which is the ratio of the

reduced duration of the contact to the oscillations period. The contact anisotropy can be achieved in

two ways: (a) by locking the active link in a specified segment of the trajectory (Table 21.5, case C, a),

and (b) by superimposing oscillations of higher frequencies (Table 21.5, case C, b), in the direction of

basic oscillations, or in the perpendicular direction of basic oscillations (Table 21.5, case C, c), normal

or tangential plane.

Piezomotors with asymmetrical oscillations are based on the asymmetry of inertia forces in non-

harmonic high frequency oscillations, multiple frequency oscillations (Table 21.5, case D, a), or forces

of dry friction with nonlinear relationship between viscosity of some liquids and velocity (Table 21.5,

case D, b). Asymmetric cycles of oscillations are generated by summing the harmonics of multiple

frequencies. The amplitude of each harmonic is chosen by using the electrode shape variations and the

area of divided electrodes or changing the amplitude of the voltage supplied. Shift in voltage supply

phases is used. The piezomotor efficiency in this case is lower, but the designs of devices are characterized

by higher up to 0.002

µm resolution in the translational drive. Besides, this leads to designing piezomotors

of limited dimensions in both coordinates, which in turn is very important in a number of applications.

Piezomotors are easily miniaturized, thus, micromotors are successfully developed. Surface acoustic

wave (SAW) motors are a good example. Piezomaterial is used as elastic substrate in SAW motor

development and, in this case, Rayleigh wave is generated in it by an interdigital transducer that comprises

two sets of interpenetrating metallic electrodes, fabricated photolithographically on the surface of the

piezoelectric substrate. Their geometry depends on the generated wave length. The driven link - slider

is arranged and preloaded on the elastic substrate so that friction force is sufficient to drive it when the

surface wave propagates. The slider has many projections of 10–50

µm in diameter in order to control

contact conditions. The SAW linear motors operate at 10 and more MHz, giving output force of several

Newtons, and the slider speed is more than 1 m/s with the positioning resolution in the order of 10 nm

and stroke in the order of 1 cm. Higher frequency of the generated wave guides to miniaturization, and

keeping the slider in contact is the main problem in it.

Advantages of piezomotors are large torque in the low speed range, wide speed range, standstill force

without excitation, high stiffness, high resolution, excellent controllability, small time constant, simple

structure and compactness, flexibility in the shape of the motor, high power/weight ratio, silent operation

since no speed reduction gears are required, and no electromagnetic induction. They give up to 1 Nm

torque and are operating at the velocity of some hundreds rpm, with resolution and time constant of

µm and ms range. Disadvantages are the requirement for wear-resistant materials and less durability due

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TABLE 21.5 Piezomotors Operating Principles

Basic Idea Schematic of Realization Remarks

Basic idea Schematic of realization Remarks

1.One active link

tuu

sin

A. Elliptic motion in the

contact: two motion

components with phase

difference

2.Two active links

tuu

sin

where u

y0,

x0,

and are

amplitudes, angular

frequency and phase of

oscillatory motions of

piezoelements

B. Elliptic motion in the

contact area: travelling

wave

ctuuu

/2cos

here u

, , and c are

amplitude, length and

velocity of wave

C. Frictional anisotropy of

contact

Usually

05.0

here

and T are the

duration of contact and

oscillation period

Wave

Output link

Input link

UyUx

Y’

)

Z, X

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Actuators 21-61

to frictional drive, suitability only for low and medium power (less than 100 W) applications, and need

of high frequency power supply.

The application of piezomotors is expanding in those markets where their specific advantages are

particularly useful. Good examples of implementation are the Canon camera’s auto focus system, and

Seiko wristwatches. New Scale Technologies’ attempts to create tiny piezomotors to achieve auto focus and

zoom optics moving in a mobile phone camera show that they meet new challenges. Piezomotors are

expected to use applications in such areas as automotive accessories, opto-mechanical systems, automatic

control and adjustment systems, robotics, military, medical, printing equipment, wearable instruments,

precision tool moving mechanisms, and so forth.

20.3.7 Piezoelectric Devices with Several Degrees of Freedom

Piezoelectric motors with several degrees of freedom have led to a new class of mechanisms, changing

their kinematic structure under control. They are based on active kinematic pair exploitation. If one or

both links of the kinematic pair are made from piezoactive material, it is possible to generate static

displacement of its elements and quasi-static or resonant oscillations, resulting in generating forces or

torque in the contact area of links. The required motion of one link relative to the other is obtained.

Because of controllable degrees of freedom, such kinematic pairs can be defined as active. An active

kinematic pair is characterized by:

•

Control of number of degrees of freedom. The simplest one is to control friction in the pair,

usually when the elements of the pair are closed by force. Here, either the friction coefficient or

the magnitude of the force executing the closure can be varied. This is achieved by excitation of

high frequency tangential or normal oscillations in the contact area of the pair.

•

Generation of forces or torque in the contact area between links. The direction of generated forces

or torque is controlled by a spacial phase shift of oscillations, for example, by activating particular

electrodes among the sectioned ones in the piezoelectric transducer.

•

Possibilities to realize additional features: self- diagnostics, multi-functionality, self-repair, and

self-adaptation.

An example of a positioning system on a plane is given in Figure 21.84. The hollow piezoceramic

cylinder with electrodes on its internal and external surfaces is situated on the plain plate. The electrodes

TABLE 21.5 Piezomotors Operating Principles (continued)

Basic Idea Schematic of Realization Remarks

D. Asymmetrical oscillations

cycles

)

N/A

1 2

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