Design and Analysis of A Recoil-Type Vibrotactile Transducer · This article describes the design of a high-bandwidth, iron-less, recoil-based electromagnetic vi-brotactile actuator

Design and Analysis of A Recoil-Type Vibrotactile Transducer

Hsin-Yun YaoTactile Labs, Saint-Bruno, Quebec, Canada

Vincent Haywarda)

upmc Univ Paris 06, Institut des Systemes Intelligents et de Robotique, 4 place Jussieu, 75005 Paris,France

(Accepted June 2010 for publication in the Journal of the Acoustical Society of America)

This article describes the design of a high-bandwidth, iron-less, recoil-based electromagnetic vi-brotactile actuator. Its working principle, the theoretical analysis, the method used to determineits transfer function, its scaling properties and its design constraints are discussed along with itsfabrication and possible improvements.

PACS numbers: 43.38.Dv, 43.66.Wv, 43.80.Jz, 43.38.Ar

I. INTRODUCTION

Modern portable communication devices are noted fortheir ever-increasing data processing and display capa-bilities. With mixed success, they strive to tap intoseveral human sensory channels: vision, audition andtouch. Comparatively, the use of the haptic channel—where movement and touch sensations are combined—inorder to exchange information is achieved today only ata very basic level, as discussed in recent surveys.1,2

The only tactile feedback signal widely used in con-sumer applications is the vibrotactile signal communi-cated to the user by inducing persistent or transient os-cillations to the enclosure or to the front plate of a hand-held device. During operation, these surfaces are nor-mally in contact with the fingers of the user, which makesthis approach very practical. If the device is in a pocket,then the large surface of the enclosure makes it possibleto radiate sufficient vibration energy for signaling an alertthrough layers of clothes or even when placed in a hand-bag. Understandably, this type of signal is easy to createand can elicit salient sensations in exchange of very littleelectrical power and given a tiny space and cost budget.

A different vibrotactile stimulation approach is em-ployed by the ‘tactor’. A tactor device is designed tostimulate a small region of skin tissue, and often operateslike an acoustic transducer. While the tactor has morethan eighty years of history since R. H. Gault noticed thatthe vibration of the membrane of a telephone earpiececaused strong cutaneous sensations,3,4 it has not beenadopted by the consumer industry. This lack of adoptionis likely to be due to the necessity to maintain direct con-tact between the radiating element and the skin throughthe use of a strap or of a garment. With the tactor, theregions of stimulation are also restricted to areas that arefree of manipulative or sensing purposes, excluding, forinstance, the fingertips and the thenar region which arecrucial during interaction with a device.

Enclosure or front plate vibration, on the other hand,

a)Electronic address: [email protected]

is almost universally used in portable phones and otherdevices because it is practical. Oscillations are almostalways obtained using vibration motors because they arevery efficient in power and space and are easy to com-mission. Their efficiency arises from their mode of op-eration which is to spin an eccentric mass attached tothe shaft of a dc motor. Their expressive capabilitiesare—by principle—restricted, as further discussed in thenext section. Despite its limitations, it is possible to usevibration motors in direct contact with the skin, makinga “one note” tactor out of a vibration motor.5

In this paper, we describe a third approach that relies,like the vibration motor, on the production of vibrationsin a whole device or plate, but which is capable, like thevoice-coil tactor or the loudspeaker, of producing pre-cisely controlled vibrations over a wide frequency band-width and dynamic range. This device operates from arecoil principle by establishing a Laplace force betweenan elastically guided slug—a permanent magnet—and ancasing that contains two coils in an open magnetic circuitarrangement.

This new device can be integrated in hand-held enclo-sures since, unlike voice-coil-driven tactors, it does notrequire a mechanical ground to operate and, unlike thevibration motor, has all the features of a transducer capa-ble of linear operation over a large range. From a tactiledisplay perspective, an analogy can be drawn with visualdisplays. Black-and-white binary displays are quite use-ful, say to display numbers, but many find color imagesmore visually appealing. They also offer new opportuni-ties for interaction designers. The transducer describedin this article achieves just that: interaction designershave total freedom over the transmitted vibration wave-form.

After a brief background discussion we describe thetransducer’s working principle and its model in termsof an equivalent electrical circuit. We then describe asystem identification procedure where each componentof the circuit is either measured or derived theoretically.The model is then employed to analyze the sensitivity ofthe complete system performance with respect to eachcomponent characteristic. Possible improvements basedon this analysis are suggested.

Recoil-Type Vibrotactile Transducer 1

II. BACKGROUND

A. Basic Vibration Motor

For purposes of comparison, let us first describe brieflythe mode of operation of vibration motors. Referring toFig. 1(a), a common design is when an eccentric massattached to shaft of a small dc motor. Sometimes, thedesign is integrated as in the model shown in the right-most example of Fig. 1(a).6

force

time

(a)

(b) (c)

FIG. 1. Vibration Motors and their typical response. (a) Var-ious models from Precision Microdrives Ltd. (b) Chirp-likeresponse of a vibration motor to a step input. (c) Reduc-ing the input amplitude by a factor five renders the motoressentially useless.

Consider the limit case when such motor is attachedto a rigid mechanical ground. When the motor spins,the interaction force with the ground depends on thesquare of the angular velocity. The vibration motorsused in consumer products typically target an angularvelocity in the range of 100–200 rotations per second.7

This method makes it possible to vibrate heavy loads pro-vided that dissipative effects do not dominate the smallpower of the motor. Given a voltage step input of mag-nitude h, the spinning velocity follows a first order re-sponse, ω(t) ∝ h[1 − exp(−t/τ)], because the vibratorobeys the general dynamic behavior of a dc motor wherethe time constant, τ , depends on the ratio of its inertiato its back-emf coefficient (if the electrical time constantis ignored). The produced centrifugal force is of the form|f | ∝ ω2. If the vibrator is mounted on a case whichis much heavier than the spinning mass, then the vibra-tion obeys h2[1 − exp(−t/τ)]2 expjh[1− exp(−t/τ)]t,where the vibration amplitude (the first factor) and thefrequency (the exponent) are linked. If we plot the re-sponse of a typical vibration motor powered at full am-plitude, we obtain a profile such as that seen in Fig. 1(b),where the ramp-up period is in the range of hundreds ofmilli-seconds. This ramp-up period causes significant de-lay and is problematic when a precise sensory outcomeis desired. As a result, these motors are not suitable forscientific experiments where the temporal properties ofthe stimulus are important.

To better appreciate the fact that there is little roomgiven to interaction designers to obtain a richer behaviorfrom such motors, Fig. 1(c) shows the response of thesame motor when the input step is five times smaller.

Not only is the vibration of very low frequency, but itsamplitude is also very low. As a result, designers haveconsidered adding additional dynamics to this kind ofvibrators,8 or have developed more sophisticated controlschemes.9 Nevertheless, the scope of these improvementsis limited.

B. Moving-Coil and Moving-Magnet Actuators

Moving-coil actuators (one model is represented inFig. 2(a)), use the same principles as electromagneticloudspeakers. They comprise a voice coil, a magnet cre-ating an annular field, and a suspension mechanism. Be-low saturation, the output force is by-and-large linearlydetermined by the input current. They can be very ac-curate and do not have bandwidth limitations other thantheir structural dynamics and the ability of the drivingamplifier to push current in an inductive load.

Compared to vibration motors, they are more expen-sive and generally larger, unless designed to operate atone single resonant frequency. In the latter case, theirresponse magnitude is very sensitive to the presence ofany dissipative term in the load which is very detrimen-tal to their role as a transducer. For haptic applications,moving-coil actuators are not very suitable. Often de-signed to move very small masses, they cannot producelarge movements at low frequencies, let alone be imbed-ded in a portable device.

(a) (b)

FIG. 2. Voice Coil Actuators. Sources: (a) Moving Coil (USAMotion), (b) Moving Magnet (H2W Technologies).

Some models are designed to operated with a fixedcoil and a moving magnet, which can result in a largeroutput force for a same volume since since a fixed coilcan be thermally connected to an external heat sink (onemodel is represented in Fig. 2(b)). The downside is alarger moving mass and often a more limited displace-ment range. Moving magnet actuators have been usedin research projects for different purposes with differentsizes and configurations. At a larger scale, permanentmagnet linear actuators are widely used in air compres-sors.10,11 At a small scale, it can be produced with sub-millimeter sizes to use as precision actuator.12,13 Becauseof its simplicity and efficiency, it has a wide spread in-dustrial and research application. Experimental and an-alytical modeling, optimization, improvements has beenextensively discussed in the past.14–16


C. Other prime mover techniques

For vibrotactile haptic applications, other actuatingprinciples have been explored: piezoelectric benders17,stacks18, and magnetostriction. None of these ap-proaches have approached the potentially low manufac-turing cost and the practicality of electromagnetic trans-ducers with either moving magnets or moving coils.

D. Design criteria for recoil-type vibrotactile actuators

The general design criteria of recoil-type vibrotactileactuators are similar to the criteria that apply to manyother types of actuators: small size, high specific power,efficiency, ability to be packaged, robustness, linearity,and ease of commissioning. However, they do have somespecific constraints. Ideally, vibrotactile actuators shouldbe able to provide stimulation from a few Hz to 500 Hz.With recoil-type actuators, providing output the upperfrequency range is not difficult, but providing output inthe low-end is constrained by a basic tradeoff driven bythe scale at which the device is built. For a given output,a reduction of the moving mass must be compensated bya larger peak-to-peak excursion, complicating the sus-pension design. This is similar to loudspeakers when thearea of the membrane is reduced. Informal experimentshave shown that 3 G of acceleration is the greatest stim-ulation at the fingertips that can be sustained before theonset of numbness. This figure was therefore targeted inthe present design.

III. ACTUATOR DESCRIPTION

A. Physical Description

The actuator, henceforth named an “haptuator”, is anungrounded moving magnet voice-coil type linear actua-tor with a moving magnet. Referring to Fig. 3, a cylin-drical magnet is suspended by two rubber membranes (ingray) via two non-ferromagnetic pins (hashed lines) in-side a tubular enclosure (hashed lines) such that the gen-erated magnetic field intersects the two halves of a coil atmostly right angle. When the current flows through thecoil, it interacts with the magnetic field of the permanentmagnet to create a Laplace force in the axial direction.The magnet moves axially as a result of this force rela-tively to the enclosure. The displacement is a functionof the Laplace force, of dynamic inertial forces actingon both the enclosure and the magnet, plus elastic andviscous forces developing between the magnet and theenclosure.

Recoil-type actuators can be small. The authors havemade extensive use of a version that is 13 mm in diam-eter, 25 mm in length, see Fig. 3(b), and yet is able toproduce several G’s of acceleration given 5 W of inputpower. Similar designs have been used in a number ofprevious studies7,19–21. Typically, the actuator was em-bedded in other enclosures or in the handles or largerdevices. In these designs, it was easy to arrange for the

N S

(a) (b)

(c) (d)

FIG. 3. Actuator arrangement. (a) Shaded parts are made ofnon-ferromagnetic material. A magnet is suspended by tworubber membranes in gray. The field lines intersect each halfof the coil to create an axial force. The coil is arranged suchthat the current flows in opposite directions in each half. (b)External aspect of a 13×25 mm ‘haptuator’. (c) Measure-ment setup with purely inertial load. (d) The magnetic fieldlines emanating from the magnet. The arrows show the forcesacting on the coil.

coil to have a 5 Ω impedance so the actuator could bepowered with ordinary audio components.

B. Dynamic Model

Denote by B(l), the magnetic flux density created bythe permanent magnet where the current i flows in thecoil at location l, the Laplace force f is obtained from

f = i

∮dl ×B(l) = γ i, (1)

where γ represents the actuator’s drive factor, or Bl fac-tor. This force, like in a loudspeaker, develops betweenthe coil and the magnet. Since the actuator is intended tooperate ungrounded, connected to driven device, its me-chanical behavior can be represented by a six-parametermodel, see Fig. 4(a).

An electromechanical force is applied to masse Mh,representing the equivalent inertia of the coil assembly,the device enclosure and the apparent hand mass, andto a second mass, Ms, representing the inertia of theslug/magnet. These elements are connected by spring-damper systems, where Kh, Bh represents the load andKs, Bs represents the suspension. The positions of themasses are denoted xs and xh, respectively.

Electrically, the system can be represented by the cir-cuit shown in Fig. 4(b) where, v is the voltage at theterminals, i, the current, ve, the voltage resulting fromLenz’ law with actuator constant ε, R and L, the resis-tance and inductance of the coil respectively.

From these diagrams, the system’s governing equations


MhMs

Bs Bh

KhKs

−ff

xhxs

+

L

i

R

v ve

(a)

(b)

FIG. 4. Actuator physics. (a) Free-body diagram of the ac-tuator. The two masses are free-floating. The ‘s’ parameterscan be set by design. The ‘h’ parameters represent the loadwhich must be driven. (b) Electrical circuit.

can be written as follows,

f = Msd2xsdt

+Bsd(xs − xh)

dt+Ks(xs − xh), (2)

−f = Mhd2xh

dt+Bh

dxhdt

+Khxh,

−Bsd(xs − xh)

dt−Ks(xs − xh), (3)

f = γi, (4)

ve = εd(xs − xh)

dt, (5)

v = Ri+ Ldi

dt− ve. (6)

C. Equivalent Circuit

A practical approach to analyze mechatronic systemsis to convert the mechanical part into an electrical equiv-alent circuit. Once connected to an electrical driver cir-cuit, the complete system can then be analyzed in thetime or in the frequency domain although it may bemade of components governed by mechanical, electrical,hydraulic, or acoustical laws. Once these interconnectedcomponents are converted into a single circuit, the con-tributions of each element can be studied using circuittheory. Such an approach has been extensively used toanalyze acoustic transducers and enclosures,22–24 and itis the method adopted in this article.

To find the equivalent circuit of a mechanical system,one can proceed with either the impedance or the mo-bility analogy. The two analysis are equivalent, and thechoice is often based on the form of the governing laws ineach domain.25 After replacing the mechanical elementsinto its electrical equivalent using the conversion Table I,each element can be connected in a way that satisfies theconservation laws. For example, using the mobility anal-ogy, if two mechanical elements are subject to the sameforce, their electrical counterparts must be connected inseries to share the same current; if they are forced tomove at the same velocity, their counterparts must beconnected in parallel in order to drop the same voltage.It must be nevertheless remembered that these analogies

are mere tools for analysis and often lead to interpreta-tions that do not possess a valid physical significance.25

TABLE I. Electrical equivalents to mechanical elements usingthe impedance or the mobility analogy.

Mechanical Impedance Mobility

Force f Voltage v: f → v Current i: f → i

Velocity x Current i: x→ i Voltage v: x→ v

Mass M Inductor L: M → L Capacitor C: M → C

Spring K Capacitor C: K → 1/C Inductor L: K → 1/L

Damper B Resistor R: B → R Resistor R: B → 1/R

From the mechanical free-body diagram, one obtainstwo equations describing the different forces acting onMs and Mh,

0 = Msdxsdt

+Bs(xs − xh) +Ks

∫(xs − xh)− f, (7)

0 = Mhdxhdt

+Bs(xh − xs) +Ks

∫(xh − xs) + f

+Bhxh +Kh

∫xh. (8)

When employing the impedance analogy an inductor Lis a substitute for a mass M , the inverse of a capacitance,1/C, for a stiffness, K, a resistance, R, for a damper, B,a current, i, for a velocity, x, such that equations thatdescribes force balances become voltage loop equations,

0 = Lsdisdt

+Rs(is − ih) +1

Cs

∫(is − ih)− v, (9)

0 = Lhdihdt

+Rs(ih − is) +1

Cs

∫(ih − is) + v

+Rhih +1

Ch

∫ih, (10)

0 = Lsdisdt

+ Lhdihdt

+Rhih +1

Ch

∫ih (11)

To obtain the electric circuit described by the theseequations, one considers them as the application of Kirch-hoff’s voltage law for two loops in a circuit. The currentin each branch is such that the three loops form a circuitshown in Fig. 5(a).

v

Ls

isCs

Ch

Lh

Rh

Rs

ih

Cs

Ch LhRh

Rs Lsi

vh

vs

0

(a) (b)

FIG. 5. Equivalent circuit. (a) Using the impedance analogy.(b) Using the mobility analogy.


Similarly, to use the mobility analogy, C is a substi-tute for M , 1/L for K, 1/R for B, voltage v for velocityx, such that the equations that describe force equilibriabecome current summations at the nodes as shown inFig. 5(b),

0 = Csdvsdt

+vs − vhRs

+1

Ls

∫(vs − vh)− i, (12)

0 = Chdvhdt

+vh − vsRs

+1

Ls

∫(vh − vs) + i

+vhRh

+1

Lh

∫vh, (13)

0 = Csdvsdt

+ Chdvhdt

+vhRh

+1

Lh

∫vh (14)

The above equations can be viewed as the result of theapplication of Kirchhoff’s current laws to three nodes of acircuit, where the current flowing through each elementis equal to the voltage at its terminals divided by itsimpedance.

Fig. 6 shows the equivalent circuits of the mechani-cal system using the impedance and mobility analogies(panel (a) and (b) respectively). These transformationsresult in circuits that are dual to each other, and solvingone is equivalent to solving the other. Note that for theimpedance analogy, the values of the impedances are theinverse of the values for the mobility analogy (Z → 1/Z)because of their duality properties. Despite the inverserelationship, all have the unit of Ohm (Ω).

v

ZhZm

Zs

Zm

Zs

Zh

i

(a) (b)

FIG. 6. Block representation of the mechanical components,(a) impedance, (b) mobility.

One important advantage of translating a mechanicalsystem into an equivalent electrical circuit is that it canbe combined with the electrical driver circuity, and thewhole mechatronic system can be analyzed in a unifiedmanner. Specifically, because audio amplification com-ponents are typically voltage amplifiers, it is more conve-nient to employ the mobility analogy. (If current ampli-fiers were used, we would prefer the impedance analogy.)Using the mobility analogy, one may combine the circuitsfrom mechanical and electrical components by mean ofan ideal transformer of ratio γ : 1 to represent the forcedeveloped between the magnet and the coil, as describedby (1) and as shown in Fig. 7(a). The relationship fromthe input voltage to the output velocity can therefore bedetermined directly. A similar approach is used in theanalysis of loudspeakers driven by voltage amplifiers.

One last transformation can be applied to the systemrepresentation to facilitate its analysis. Fig. 7(b) showsa circuit equivalent to that of Fig. 7(a) without the ideal

Zc

γxm xm

f

v

1/γ Zm

Zs

Zh

fZm

Zs

Zh

v

γ

Zc

γ2

xh

(a)

(b)

FIG. 7. Circuit representation of the electrical circuit com-bined with the equivalent circuit of the mechanical system.

transformer, by reflecting the primary side into the sec-ondary side. Solving for the circuit variables now allowsus to establish the relationship between the input v andother variables of interest, such as xh.

These transformations can be summarized by expres-sions in the Laplace domain:

Zc = sL+R, (15)

Zm =1

sCm=

1

sMm, (16)

Zs =

(1

sLs+

1

Rs

)−1

=

(Ks

s+Bs

)−1

, (17)

Zh =

(sCh+

1

sLh+

1

Rh

)−1

=

(sMh+

Kh

s+Bh

)−1

. (18)

IV. SYSTEM IDENTIFICATION

Some parameters such as the mass of the magnet orof the shell, mm and mh, as well as input voltage canreadily be known or measured. Some others, such asthe mechanical impedance of the suspension Zs, the coilimpedance Zc, and the actuator constant γ require ex-perimental identification.

A. System Response and Modeling

In a first step the input-output system response wasmeasured when the load was just an inertia, that is,when Kh = 0 and Bh = 0. To achieve this condition,we secured the haptuator in a box and suspended it withthin threads. An accelerometer (adxl320 from AnalogDevices) was attached to the wall of the box. When ac-tivated, the actuator could vibrate freely in the directionparallel to the floor, see Fig. 3(c).

The acceleration response for three different input volt-ages can be see in Fig. 8(a). The three responses for 1,2 and 3 V inputs (rms) do not fall exactly in top ofeach other (by less than 1 dB). This slight discrepancyis easily explained by saturation effects which are mostlynoticeable around the natural resonance. They arise, like


for loudspeakers, because the large excursion of the mov-ing part causes the suspension to cease acting linearly,presumably because of the nonlinear characteristics ofrubber at high strains, and because the magnetic flux in-terfering with the coils is no longer constant. In otherwords, the actuator starts “bottoming out” at resonancefor a 3 V input. For this input, the acceleration is by-and-large 3 G throughout the frequency range. For a 1 Vinput (1 G output for this load), the actuator operatesmostly linearly.

The shape of the response resembles that of a 2nd ordersystem with a 60 Hz natural resonance, which decaysfollowing one slope up to around 150 Hz, and followinganother from 150 Hz to 500 Hz, indicating the presenceof zeros in the transfer function. A 2nd order model withtwo poles and two zeros gave an excellent fit (r2 = 0.98).It is written

H(s) =xhv

=0.685s(s− 1150)

s2 + 565.9s+ 1.283e5, (19)

=as(s− b)

s2 + 2ζ wo s+ w2o

.

where the damping coefficient is ζ = 0.79 and the reso-nance frequency F0 = ω0/2π = 57 Hz. It is not seen onthe figure, but the model gradually differs from the mea-sured response for frequencies above 400 Hz, indicatingthe possibility of one or several zeros at higher frequen-cies. These zeros are likely to be due to the accelerometertransfer function (rated bandwidth 500 Hz), but not tothe actuator.

B. Parameters found directly

Due to the small number of turns, the inductance of thecoil L (found to be 2.05×10−8 H) could be neglected inthe frequency range considered, hence Zc could be mea-sured directly to be Zc = 5.0 Ω. Other parameters di-rectly measurable are the mass of the magnet, Mm = 7 g,and the mass of the case and of the test box, Mh = 15 g.

The actuator drive factor, γ, (or ‘Bl’ factor) was ob-tained using two methods: force measurement and sim-ulation.

The first method involved measuring directly the forceproduced by a dc current. The rubber suspension wasremoved, and a peg was used to keep the magnet cen-tered. The actuator was placed vertically above a scalesuch that when a dc current was sent to the coil, themagnet produced a force acting against the scale. Hav-ing the force and the current, using equation (1), γ canbe obtained.

The second method employed a commercial finite ele-ment analysis tool (comsol 3.5, comsol ab, Stockholm,Sweden). The geometry of the actuator was supplied tothe simulation, as well as the B field of the magnet, mea-sured to be 0.50± 0.01 T at the pole, and the total coillength. The software simulation tool used the above pa-rameters to estimate γ. A plot of the magnetic field linesand the resulting Lorentz force from the simulation canbe seen in Fig. 3(d).

Twelve measurements were made with i=70–143 mA,using the first method. The factor γ was foundto be 0.98±0.06 T·m. The second method gaveγ=0.96±0.01 T·m. These values are within each other’suncertainty range. In the following sections, the meanvalue of γ = 0.97 T·m is used.

C. Suspension Impedance

From Fig. 7(a) the transfer function from voltage tooutput velocity in terms of the system parameters is

xhv

=γZsZh

γ2ZsZm + Zc(Zm + Zh + Zs),

H(s) =s γZsZh

γ2ZsZm + Zc(Zm + Zh + Zs). (20)

The quantities Zs, Zc, Zm, and Zh can be substitutedin (20) for their expressions, (15)–(18), to give

H(s) =s γZsZh

γ2ZsZm + Zc(Zm + Zh + Zs)

=s γ(Ks/s+Bs)

−1(sMh)−1

γ2(Ks/s+Bs)−1(sMm)−1 +R((sMm)−1 + (sMh)−1

+ (Ks/s+Bs)−1)

=s2γMm

s2RMmMh + s(γ2Mh +RMhBs +RMmBs)(21)

+RMhKs +RMmKs.

The transfer function (21) is of 2nd order with two zerosand two poles with two unknown parameters, Ks and Bs

that describe a 1st-order suspension model as in (17).Substituting in (20) the values Mm=0.007, Mh=0.015,γ=0.97, and R=5.0, the transfer function becomes

H(s) =s γZsMm

Zsγ2Mh +RMh +RMm + sRZsMmMh

=64.7sZs

134Zs + 1.05e3 + 5.0sZs. (22)

An expression for Zs can be found by relating (22) withthe transfer function (19) obtained experimentally,

Zs(s) =11.7 (s− 1150)

s2 + 659s+ 1.37e05. (23)

Instead of a 1st-order expression, Z(s) = (Ks/s+Bs)−1

from (17), we find a more complicated 2nd-order expres-sion. This can easily be explained by considering thatthe disk suspension operates as a vibrating membrane.Its actual dynamics are likely to involve high-frequencyvibrations modes which were picked up by the experi-mental identification procedure in the frequency range ofinterest.

The denominator of (22) suggests that the first term,134Zs, is the dominant term for position of the poles.When comparing the response of Zs in (23) (see plotin Fig. 8(c)) to the model (19), it is noticed that theyshare the same zeros at 0 and 1 150 rad/s, and that they


frequency (Hz)50 100 200 300 400 500

1.0

gain

(rat

io)

1.5

0.5

0

4

2

0

-2

-4

200

gain

(dB)

300

100

0

phas

e (d

eg)

3 V2 V

1 V

frequency (Hz)50 100 200 300 400 500

-20

-30

-40

-50

50

-50

gain

(dB)

100

-100

0

phas

e (d

eg)

(a)

(b) (c)

FIG. 8. System Response. (a) Frequency response of the actuator acceleration to input voltage ratio for three different levels(rms). (b) The magnitude and phase plot of a fitted 2nd order model. (c) Plot of the impedance of the suspension, Zs.

have poles that are very close to each other (358 vs. 370rad/s). The resemblance of the transfer function with thesuspension impedance suggests that the transfer functionis primarily determined by the characteristics of the sus-pension and shaped by the other parameters.

V. ACTUATOR ANALYSIS

With the help of the parametrized transfer func-tion (20), we can now investigate the impact of each in-dividual parameter on the overall outcome. The effectof each element in (20) is analyzed, except the magnetmass Zm. The magnet plays a dominant role in the di-mensioning of the actuator, and changing it would meanmodifying the whole geometry of the actuator. For thisreason, the magnet mass Zm is taken to be the scale pa-rameter.

Figure 9 shows the effect of scaling by varying eitherthe radius or the length of the cylindrical magnet. Allother parameters are kept constant, except γ, which wasre-evaluated for each case. The figure shows an almostlinear relationship between the relative mass of the mag-net and the magnitude of the output ratio at 100 Hz.The shape of the frequency response for all cases is al-most identical (not shown). There is a slight advantageat scaling radius rather than length. This may be dueto the fact that a magnet with a larger radius providesa greater surface for the field lines and therefore gives alarger flux at the location of the coils.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.40.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

mass ratio

outp

ut ra

tio a

t 100

Hz

radius scalinglength scaling

FIG. 9. Actuator scaling properties.

A. Impedance of the suspension: Zs

As noted in the previous section, Zs dominates theplacement of the system zeros and poles. Physically, thesuspension of the actuator is a thin rubber disk designedto be much more compliant in the axial direction than inthe radial direction. At high frequencies it behaves like amembrane with vibration modes that gave Zs a 2nd-ordertransfer function characteristic (23).

The modification of the material and of the thicknessof the suspension would change the acceleration responseof the actuator according to (22). Figure 10 shows thevariation of H(s) when the real (top) and imaginary (bot-tom) parts of Zs vary by±20% around the nominal value.Varying the real part of Zs does not have the same effectas varying the imaginary part but both variations shift


the resonance and modify its Q-factor.

frequency (Hz)50 100 200 250 300 350 400 450 500150

Im(Zs)

Re(Zs)5

0

-55

0

-5

mag

. (dB

)m

ag. (

dB)

FIG. 10. The magnitude of H(s) when the real (top) andthe imaginary part (bottom) of Zs vary by 20% around thenominal value.

B. Load Inertia Zh

The primary function of this actuator is to providea source of vibration. Figure 11(a) shows how differ-ent inertia affect the acceleration output, based on (20).As might be expected, the acceleration output decreaseswith a larger load. The phase plot, however, is unaf-fected when Zh varies from 15 to 100 g in 5 g increments.The higher the mass, the denser the lines, which meansthat the magnitude doesn’t decrease linearly with the in-creasing mass. The higher the mass, the smaller is theincremental effect on the output decrement.

C. Actuator drive factor γ

The larger is the B field, the larger is γ. Figure 11(b)shows how a higher value of γ enhances the response.Similarly to Zh or Zc, the output magnitude is almostdirectly proportional to the increase of γ, while the phasebehavior remains unchanged. The software analysis toolwas also used to investigate different strategies to in-crease γ without changing significantly the geometry, thesize or the electrical characteristics of the haptuator.

1. Coil Placement

The coil of the actuator should be placed where themagnetic field is the strongest, which is when the mid-point of the coil is near with edge of the magnet with asmall offset, see Fig. 12(a). Because the magnetic fieldis not completely symmetrical, moving the coil sightlytoward the denser area increases the output force. Theoptimal placement is when the mid-point of the coil isoffset by about 0.4 mm with respect to the edge of themagnet.

2. Coil Configuration

Another possible optimization is to attempt pack asmany turns as possible where the magnetic field is

5

mag

nitu

de (d

B)ph

ase

(deg

ree)

0

-5

-10

-15

-20

150

100

50

5

10

frequency (Hz)

mag

. (dB

)ph

ase

(deg

ree)

0

-5

150

100

50

50 100 200 250 300 350 400 450 500150

(a)

(b)

FIG. 11. (a) Acceleration output when the load varies from15 g to 100 g (lowest) with 5 g increments. The thick line inthe magnitude plot with the nominal value of 15 g. (b) Ac-celeration output for values of γ 30% higher to 30% lower.Each line represents a 10% increment. The thick line in themagnitude plot with the original value, γ=0.97.

0.957

1.124

1.2161.235

2x40

3x27

4x20 5x16

0 0.2 0.4 0.6 0.80.9520.9540.9560.958

N S

coil position from the center (mm)

(a)

(b)

FIG. 12. Coil optimization. (a) Values of γ for different coilpositions away from the center position. (b) Actuator drivefactor γ given by different coil configurations.

strongest. This can be achieved by shortening the coillength and increasing the number of layers, while keep-ing the same total number of turns. A tradeoff existssince the field decays radially. Simulations demonstratedthat γ improves with the number of layers until the coilthickness becomes large enough to prevent any further


improvement, see Fig. 12(b). A thicker coil is also heav-ier.

D. Summary of Performance Figures

The basic design described in this paper is likely tobenefit from further optimizations and improvements.Table II summarizes the basic performance figures of thepresent prototype for future reference.

TABLE II. Summary of the main characteristic figures.

Parameter Value

Dimensions (diameter × length) 14 × 29 mm

Total mass 23 g

Mass of moving magnet 7 g

Drive factor 0.98 T·mAcceleration 3V input, 125 Hz 3.0 G (29.4 m/s2)

Rated Bandwidth 50–500 Hz

Typical Impedance 6.0 Ω

Maximum Input Voltage 3.0 V

Maximum Input Current 0.5 A

VI. CONCLUSION

The goal of this paper was to describe, model, andanalyze a new type of iron-less recoil actuator. Once amathematical model of the actuator was obtained, thecontribution of each individual components could thenbe studied separately. We started from a mechanicalfree-body diagram and then translated it into a equiv-alent of electrical circuits so that it could be combinedwith the electrical driving circuitry. We then found thecomplete system transfer function. While most param-eters entering in the coefficients of the transfer functioncould be measured directly, the suspension and the ac-tuator drive factor γ were identified experimentally. Itwas verified that γ could be accurately predicted by sim-ulation with finite element analysis tool. By substitutingthe experimental data into the theoretical equation, it isthen possible to find the remaining unknown suspensionmodel.

We then analyzed the sensitivity of each parameteron the overall response. We found that the suspensioncharacteristics impacts the general shape of the responseas well as the resonance frequency. In that the haptuatoris similar to a loudspeaker. The other components havea direct, linear scaling effect on performance. Increasingγ increases the output proportionally. Moreover, as canexpected, output acceleration decreases with the loads.The actuator also exhibits a simple scaling law. A finiteelement element analysis tool was shown to be able topredict the behavior of the actuator accurately and thuscan be used to reliably optimize the overall coil/magnetgeometry. Possible improvements were also discussed.

Acknowledgments

This work was funded in part by the Natural Sciencesand Engineering Council of Canada in the form of a Dis-covery Grant to VH.

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