Research Article Torque for an Inertial Piezoelectric ...downloads.hindawi.com/journals/tswj/2013/854126.pdf · e Scientic World Journal · Housing Adapting piece Bolt Wire Needle

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 854126 9 pageshttpdxdoiorg1011552013854126

Research ArticleTorque for an Inertial Piezoelectric Rotary Motor

Jichun Xing and Lizhong Xu

Mechanical Engineering Institute Yanshan University Qinhuangdao 066004 China

Correspondence should be addressed to Lizhong Xu xlzysueducn

Received 26 September 2013 Accepted 24 November 2013

Academic Editors G Carbone and O D Makinde

Copyright copy 2013 J Xing and L Xu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For a novel inertial piezoelectric rotary motor the equation of the strain energy in the piezoceramic bimorph and the equations ofthe strain energy and the kinetic energy in the rotor are given Based on them the dynamic equation of themotor is obtained Usingthese equations the inertial driving torque of the motor is investigatedThe results show that the impulsive driving torque changeswith changing peak voltage of the excitation signal the piezoelectric stress constant the thickness of the piezoceramic bimorphand the rotor radius obviously Tests about the motor torque are completed which verifies the theory analysis here in The resultscan be used to design the operating performance of the motor

1 Introduction

Piezoelectric motors maintain relatively high torque at rela-tively low speeds without a reduction gear [1 2] Among allof the piezoelectric motors inertial drive principle has theadvantage of simpler requirements to the construction anddriving circuitry [3]

One type of the inertial motors is based on the impactdrive mechanism from impulse inertial force [4] Using themechanism the micromanipulator for cell manipulation andauxiliary positioning system for STM and AFM were devel-oped [5 6] Another type of the inertialmotors is based on thesmooth impact drive mechanism Here a base plate or bar isdriven with rapid expansion and slow shrinkage The slideron the base slips during rapid motion and follows the basedue to frictional force With this principle many applicationswere proposed and fabricated [7ndash9] Choi et al proposed adynamic model to investigate dynamic characteristics of anovel type of inertial actuator and verified themodel throughcomparison of voltage-dependent actuating forces betweenexperiment and analysis [10] Lipanov et al proposed aninertial piezoelectric step drive for subnanosize-accuracymovements [11] Gulyaev et al used backlash-free screw-nutpairs in an inertial piezodrive to increasemovement accuracyof an inertial piezoelectric drive [12] Mazeika and Vasiljevproposed a novel design of tiny piezoelectric inertial motorbased on inertial motion of the slider applying sticks andslip phases between stator and slider and a prototype inertial

piezoelectric motor was built The motor has simple designand consists of the slider with a bimorph piezoceramic discand the clamped cylindrical shaft used for sliding [13]

However these motors belong to the linear inertial pie-zoelectric motors and the inertial piezoelectric rotary motorwas seldom reported For further miniaturization of therotational piezoelectric motors the authors proposed a novelinertial piezoelectric rotary motor [14] The motor structureis simple and further miniaturization of the rotational piezo-electric motor can be done easily

However the inertial driving torque of the motor has notbeen investigated yet It is unfavorable to design the load-carrying ability of the motor

In this paper the equations of the strain energy in thepiezoceramic bimorph and the equations of the strain energyand the kinetic energy in the rotor are deduced Based onthem the dynamic equation of the motor is obtained Usingthese equations the inertial driving torque of the motor isinvestigated Tests about the relationship between the motortorque and the exciting voltage frequency or amplitude aredone The results are useful for design and control of theoperating performance of the motor

2 Voltage Excitation

As shown in Figure 1 the novel inertial rotary motor consistsof a stator and a rotor The rotor includes an outer ring

2 The Scientific World Journal

middotHousing

Adapting piece

Bolt

Wire

NeedlePiezoelectricityceramic piece

Signalsource

Vibrator

Shaft

Base

(a) Motor structure

BoltNeedle

Vibrator

Base

BearingShaft

Adapting pieceHousing

(b) 3D model

Pzt minus

minus

+

+

(c) Rotor structure and piezoceramic bimorph arrangement

Figure 1 Motor and rotor structures

an inner ring and two ribs connecting the outer ring tothe inner ring The inner ring is mounted on a supportingbearing and the outer ring is used as the inertial mass Thepiezoceramic bimorph is adhered on each side surface of thetwo ribs As soon as a rapid rise input voltage is suppliedto the motor it excites the transverse bending vibration ofthe two ribs Thus the inertial force within the outer ringoccurs which causes inertial torque to be applied to the rotorand makes it rotate Then a slow decreasing input voltage issupplied to the motor so the inertial forces within the outerring are so small that the inertial torque can be balancedby the friction torque between the rotor and the bearingThus the rotor does not rotate The rapidly increasing andslowly decreasing input voltage with a special frequency issupplied to the motor periodically and then the rotor canrotate continuously

For the motor motion a saw-tooth-type electric signal isapplied (see Figure 2) Here 119905 is the time 119879 is the period ofthe saw-tooth cycle 120583 (0 lt 120583 lt 1) is the ratio of the rise timeto the period and 119860 is the amplitude of voltage signal 119881(119905)The voltage signal can be written by

119881 (119905) =

119860

120583119879119905 (0 le 119905 lt 120583119879)

119860

(120583 minus 1) 119879119905 +

119860

1 minus 120583(120583119879 le 119905 le 119879)

(1)

The voltage signal can be written in the Fourier series as

119881 (119905) = 1198860+

infin

sum

119899=1

[119886119899cos (119899120596119905) + 119887

119899sin (119899120596119905)] (2)

where

1198860=

1

119879int

119879

0

119881 (119905) 119889119905

=1

119879(int

120583119879

0

119860

120583119879119905119889119905 + int

119879

120583119879

(119860

(120583 minus 1) 119879119905 +

119860

1 minus 120583)119889119905)

=1

2119860

119886119899=

2

119879int

119879

0

119881 (119905) cos (119899120596119905) 119889119905 =119860 (minus1 + cos (2119899120583120587))

211989921205872120583 (1 minus 120583)

119887119899=

2

119879int

119879

0

119881 (119905) sin (119899120596119905) 119889119905 =119860 sin (2119899120583120587)

211989921205872120583 (1 minus 120583)

(3)

3 The Impulsive Moment

The rotor of the motor can be considered as a beam withinertial mass at its two ends Four piezoceramic bimorphs are

The Scientific World Journal 3

V(t) A

T120583T

t

0

Figure 2 Applied electrical signal wave

y

x mm

Pzt

ll

Figure 3 Dynamics model of the rotor

adhered on side surface of the beam (see Figure 3)The strainin the piezoceramic bimorphs can be calculated as

1205761=

ℎ

2

1205972119910 (119909 119905)

1205971199092

(4)

where 119910(119909 119905) is the transverse displacement of the beam 119909 isthe length coordinate of the beam ℎ is the thickness of thebeam and 120576

1is the strain in the piezoceramic bimorphs

By substituting (4) into the piezoelectric equation thestress in the piezoceramic bimorph can be given [15] as

1205901 (119909 119905) = minus119890

311198643+

ℎ

2119888119864

11

1205972119910 (119909 119905)

1205971199092

(5)

where 1198643is the electric-field intensity on the piezoceramic

bimorph 1198643

= 119881(119905)ℎ119901 ℎ119901is the thickness of the piezoce-

ramic bimorph 11989031

is the piezoelectric stress constant 11988811986411

isthe stiffness constant and 120590

1(119909 119905) is the stress in the piezo-

ceramic bimorphFrom (4) and (5) we know that the strain energy 119881

119901in

the piezoceramic bimorph is

119881119901=

119887

2int

119897119901

0

1205901 (119909 119905) 1205761119889119909

=119887ℎ

4

infin

sum

119894=1

119902119894(119905) int

119897119901

0

minus11989031119864312060110158401015840(119909) 119889119909 +

1

2

infin

sum

119894=1

119896119901

119894119895119902119894(119905) 119902119895(119905)

(6)

where 119910(119909 119905) = suminfin

119899=1120601119899(119909)119902119899(119905) 119896119901119894119895

= 119896119901

119895119894= int119897119901

0(119887ℎ24)119888119864

1112060110158401015840

119894

(119909)12060110158401015840

119895(119909)119889119909 120601

119895(119909) is the mode function and 119897

119901and 119887 are the

length and width of the piezoceramic bimorph respectivelyThe strain energy 119881

119871119875in the rotor beam is

119881119871119875

=1

2int

119897

0

119864119868[1205972119910 (119909 119905)

1205971199092

]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119896119897

119894119895119902119894 (119905) 119902119895 (119905)

(7)

where 119896119897

119894119895= 119896119897

119895119894= int119897

minus11989711986411986812060110158401015840

119894(119909)12060110158401015840

119895(119909)119889119909 and 119897 is the length of

the rotor beam (see Figure 3)Four piezoceramic bimorphs are used in the motor and

then the total strain energy 119881 in the rotor and the piezo-ceramic bimorphs is

119881 = 4119881119901+ 119881119871119875

(8)

Kinetic energy 119864119898of the rotor is

119864119898

=1

2int

119897

minus119897

120588119878[120597119910

120597119905(119909 119905)]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119898119894119895

119902119894(119905) 119902119895(119905)

(9)

where119898119894119895= 119898119895119894

= int119897

minus119897120588119878120601119894(119909)120601119895(119909)119889119909 +119898120601

119894(minus119897)120601119895(minus119897) +119898120601

119894

(119897)120601119895(119897) is the equivalentmass of the rotor and119898 is the inertial

massSubstituting (8) and (9) into Lagrange equation yields

M 119902 (119905) + C 119902 (119905) + K = F (119905) (10)

where M = [119898119894119895] is the mass matrix K = [119896

119894119895] is the stiffn-

ess matrix 119896119894119895

= 119896119901

119894119895+ 119896119897

119894119895 F(119905) is the generalized force vec-

tor C = [119888119894119895] is the damping matrix and 119888

119894119895= 119888119895119894

= int119897

0119862119889120601119894

(119909)120601119895(119909)119889119909 here the friction damping between stator and

rotor is considered and the equivalent damping coefficientsare determined from equal energy principle

Using the orthogonality of the mode functions (10) canbe changed into the following form uncoupled to each other

Mn 119902 (119905) + Cn 119902 (119905) + Kn119902 (119905) = Fn (119905) (11)

where Mn Cn and Kn are the diagonal mass damping andstiffness matrixes Fn(119905) = 1198651(119905) sdot sdot sdot 119865

119894(119905) sdot sdot sdot

119879 is the regu-lar force vector

Each element 119865119894(119905) of the regular force vector is

119865119894 (119905) =

119887ℎ

4(2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119886) 12060110158401015840

119894(119909) 119889119909

+2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119887) 12060110158401015840

119894(119909) 119889119909)

=119887ℎ11989031119881

2ℎ119901

(12060110158401015840

119894(119909119886) + 12060110158401015840

119894(119909119887))

(12)

where 119909119886and 119909119887are the average positions of the piezoceramic

bimorphsThe solution of (11) is

119902119895=

1

119872119895120596119903119895

int

119905

0

119865119895(120591) 119890minus120585119895120596119895(119905minus120591) sin120596

119903119895(119905 minus 120591) 119889120591 (13)


times10minus6

times10minus6

2

1

0

minus10 05 1 15

0 05 1 15

times10minus3

times10minus3

t (s)

t (s)

y(m

)T

(Nm

) 5

0

minus5

(a) 120596 = 10000 rads

t (s)

t (s)

y(m

)T

(Nm

)

times10minus7

times10minus7

times10minus4

times10minus4

5

0

minus5

15

10

5

0

0 2 4 6 8

0 2 4 6 8

(b) 120596 = 20000 rads

Figure 4 The displacement response and the corresponding impulsive moment

By substituting (2) and (12) into (13) and neglecting thetransient solution the steady solution can be obtained

119902119895=

119865119899

1198721198951205962

119895

1198860+

infin

sum

119899=1

1

radic(1 minus 1205742

119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895) + 119887119899sin (119899120596119905 minus 120593

119899119895)]

(14)

Here 120593119899119895

= arctan(2120585119895120574119899119895(1 minus 120574

2

119899119895))

The steady response of the rotor to electric excitation is

119910 (119909 119905) =

infin

sum

119895=1

120601119895 (119909) 119902119895 (119905)

=

infin

sum

119895=1

119865119899120601119895 (119909)

1198721198951205962

119895

times

1198860+

infin

sum

119899=1

radic1198862

119899+ 1198872

119899


119899119895)2

+ (2120585119895120574119899119895)2

times cos(119899120596119905 minus 120593119899119895

minus arctan(119887119899

119886119899

))

(15)

From (15) the velocity 119910(119909 119905) and acceleration 119910(119909 119905) ofthe motor rotor can be given

0 05 1 15 25 3 350

1

2

3

4

2

0

5

10

15

Displacement of the endInertial moment of the end

times10minus6times10minus6

minus5

times104

y(m

)

(rads)

T(N

m)

Figure 5The frequency response of the displacement and impulsivemoment

From 119910(119909 119905) and 119879(119905) = minus119869ℎ120572 the impulsive moment for

the motor can be given as

119879 (119905) =

infin

sum

119895=1

119869ℎ119865119899120601119895(119897)

1198971198721198951205962

119895

infin

sum

119899=1

11989921205962


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895)

+119887119899sin (119899120596119905 minus 120593

119899119895)]

(16)

where 119869ℎis the rotary inertia of the rotor 120572 is the angular

acceleration of the rotor end and 120572 = 119910(119909 119905)119897


times10minus6

times104(rads)

T(N

m)

minus05

minus1

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

05

1

15

(a) Mode 1

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(b) Mode 2

times10minus6

minus5

times104(rads)

T(N

m)

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

5

10

(c) Mode 3

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(d) Mode 4

Figure 6 The response of the impulsive moment as a function of 120583

4 Simulation and Test

Equations in this paper are utilized for the impulsivemomentanalysis of the inertial piezoelectric rotarymotorThe param-eters of the numerical example are shown in Table 1 Thedisplacement response of the rotor end to the excitationsignals and the corresponding impulsive moment under thedifferent exciting frequencies are given in Figure 4 (here 120596is the frequency of the excitation signals) From Figure 4 thefollowing observations were worth noting

(1) Under the periodic voltage excitation with saw-toothwave the dynamic displacement at the end of therotor changes periodically The rise edge of the exc-itation voltage corresponds to multipeaks of the

Table 1 Parameters of the numerical example

1198771(mm) 119877

2(mm) ℎ (mm) 119897 (mm) 119864 (Gpa)

12 13 1 12 120119897119901(mm) 119887 (mm) ℎ

119901(mm) 119890

31(mm) 120588 (kgsdotmminus3)

10 5 05 minus0046 89 times 103

dynamic displacement which are different from eachother and its trailing edge corresponds to a large peakin an opposite direction

(2) Under the periodic voltage excitation with saw-toothwave the changes of the impulsive moment along


2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

hp = 05mmhp = 08mmhp = 1mm

(d) ℎ119901 change

Figure 7 Changes of the impulsive moment along with the system parameters

with time are similar to those of the dynamic displa-cement at the end of the rotor As the frequency ofthe excitation signal grows the peak number of theimpulsive moment drops for the rise edge of the exci-tation voltage

(3) Thedriving torque of themotor is equal to the integralof the impulsive moment at a given time range Theresults show that the integral is not zero which meansthat a driving torque is produced

(4) At a given time range the integral of the impulsivemoment is different for the different voltage excitationfrequency So the changes of the driving torque of

the motor along with the voltage excitation frequencyshould be investigated further

The frequency responses of the displacement at the rotorend and the corresponding impulsive moment are given inFigure 5 (here 120583 = 07 and 120585 = 01) Figure 5 shows thefollowing

(1) As the frequency of the voltage excitation is near one-nth of the natural frequency the peaks of the displa-cement at the rotor end and the corresponding impul-sive moment occur

(2) As the frequency of the voltage excitation is near thenatural frequency of the motor (119899 = 1) the peaks


5 10 15

0

5

10

15

20

Structure 1Structure 2Structure 3

minus5

(rads)

T(N

m)

times104

times10minus4

Figure 8 Changes of the impulsive moment along with the motorsizes

of both displacement at the rotor end and the corre-sponding impulsive moment are the maximum

(3) As the frequency of the voltage excitation is equal toone-nth of the natural frequency the jumping frompositive peak to negative peak occurs This is notfavorable to operation of the motor The frequencyof the voltage excitation should be taken as slightlysmaller than one-nth of the natural frequency

Figure 6 shows the impulsivemoment as a function of theratio of the rise time to the period for the excitation voltageIt shows the following

(1) As the ratio of the rise time to the period for the exci-tation voltage grows the peak of the impulsive mom-ent grows The effects of the 120583 value on the positiveimpulsive moment peak are relatively small and itseffects on the negative impulsive moment peak arerelatively large

(2) For the different modes the impulsive moment peaksare different from each other at the same 120583 For mode3 the impulsive moment peak is the maximum Itshows that mode 3 is the most favorable for the motoroperation

Changes of the impulsive moment along with the systemparameters are investigated (see Figure 7) Changes of theimpulsive moment along with the motor sizes are givenin Figure 8 Table 2 shows three different motor sizes andcorresponding exciting frequencies Here only the results formode 3 are given They show the following

(1) As the damping coefficient 120585 drops the impulsivemoment of the motor grows When the dampingcoefficient 120585 is reduced to one-eighth the impulsive

Figure 9 The prototype motor and its driving system

Table 2 Different motor sizes and exciting frequencies

Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336

moment is increased by more than 6 times So thesmall damping should be maintained to increase thedriving torque of the motor

(2) As the peak voltage 119860 of the excitation signal growsthe impulsivemoment of themotor growsThe impu-lsivemoment increases nearly linearlywith increasingthe peak voltage 119860

(3) As the piezoelectric stress constant 11989031

grows theimpulsive moment of the motor growsThe impulsivemoment increases nearly linearly with increasing thepiezoelectric stress constant 119890

31as well So the piezo-

electric material with larger constant 11989031

should beselected

(4) As the thickness ℎ119901of the piezoceramic bimorph

grows the impulsive moment of the motor growsSo larger thickness ℎ

119901of the piezoceramic bimorph

should be taken

(5) As themotor size grows the impulsivemoment of themotor grows obviously It is because the rotary inertiaof the motor grows obviously with increasing themotor size Hence the design of themotor size shouldbe done according to load-carrying requirement

Tests are done on the inertial rotary motor to obtainthe relationship between the motor torque or speed and theexciting voltage frequency or amplitude The motor was dri-ven by a signal generator (YB-1602) and a power amplifier(HFVA-42) Here the saw-tooth wave voltage signals wereused (120583 = 09) The prototype motor and its driving systemare given in Figure 9 The experimental results are given inFigure 10 The comparison between the experimental resultsand the calculated values is given in Table 3 They show thefollowing


6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes

Figure 10 Changes of the motor speed and torque along with exciting frequency and voltage

Table 3 Comparison of experimental results and theoretical values

Signalvoltage (V)

Calculatedmaximumtorque(Nmm)

Calculatedaveragetorque(Nmm)

Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40

(1) As the peak of the voltage signal is 100V the modelmotor starts to rotate at 120596 = 6280Hz As the exci-ting frequency grows the rotating speed of the motorincreases significantly It gets to the maximum value(596 rpm) at 120596 = 6310Hz As the exciting freque-ncy further grows the rotating speed of the motordecreases quickly It stops to rotate at 120596 = 6330Hz

(2) For 120596 = 6310Hz the motor starts to rotate at 119881 =

25V The rotating speed of the motor grows nearlylinearly with increasing the amplitude of the excitingvoltage (from 25V to 170V)

(3) The above results are obtained under the conditionwithout the outer load Here the motor only bearsthe friction force within the motor Under the outerload and 119881 = 100V the output torque of the motorincreases significantly with increasing the excitingfrequency It gets to the maximum value (035Nmm)at 120596 = 6310Hz As the exciting frequency furthergrows the torque of the motor decreases quickly Thetorque is zero at 120596 = 6330Hz

(4) For 120596 = 6310Hz The output torque of the motorgrows obviously with increasing the amplitude of theexciting voltage At 119881 = 175V the output torque ofthe motor is 052Mmm

(5) For 120596 = 6310Hz and 119881 = 75ndash175V the relative err-ors between the measured torque and the calculated


average torque are about 30ndash40 It shows that thetheory analysis results in this paper are believable

5 Conclusions

In this paper the dynamic equation of the inertial piezo-electric rotary motor is obtained The inertial driving torqueof the motor is analyzed and tested The results show thefollowing

(1) The jumping from positive peak to negative peakoccurs when the frequency of the voltage excitation isequal to the natural frequency which is not favorablefor the motor So the frequency of the voltage exci-tation should be taken as slightly smaller than thenatural frequency

(2) The impulsive driving torque changes with changingthe system parameters Larger peak voltage of the exc-itation signal larger piezoelectric stress constant lar-ger thickness of the piezoceramic bimorph and largerradius of the rotor can give larger impulsive drivingtorque

(3) The experimental output torque verifies the theoryanalysis here in

Acknowledgment

This project is supported by the National Natural ScienceFoundation of China (51275441)

References

[1] Z Boumous S Belkhiat and F Z Kebbab ldquoEffect of shearingdeformation on the transient response of a traveling waveultrasonic motorrdquo Sensors and Actuators A vol 150 no 2 pp243ndash250 2009

[2] S Pirrotta R Sinatra and AMeschini ldquoEvaluation of the effectof preload force on resonance frequencies for a traveling waveultrasonic motorrdquo IEEE Transactions on Ultrasonics Ferroele-ctrics and Frequency Control vol 53 no 4 pp 746ndash752 2006

[3] K Spanner ldquoSurvey of the various operating principles of ultra-sonic piezomotorsrdquo in Proceedings of the International Confer-ence on Actuator 2006

[4] T Morita ldquoMiniature piezoelectric motorsrdquo Sensors and Actua-tors A vol 103 no 3 pp 291ndash300 2003

[5] K Hideki H Yasuo K Hitoshi and H Toshiro ldquoPositioningand measurement using a crystalline lattice reference scalemdashafeasibility study of STM application toMEMSrdquo in Proceedings ofthe IEEE Micro Electro Mechanical Systems An Investigation ofMicro Structures Sensors Actuators Machines and Robots pp197ndash202 February 1990

[6] H Kawakatsu Y Hoshi H Kitano and T Higuchi ldquoCrystallinelattice for metrology and positioning controlrdquo in Proceedings ofthe IEEEMicro ElectroMechanical Systems (MEMS rsquo91) pp 239ndash244 February 1991

[7] A Bergander J-M Breguet C Schmitt andR Clavel ldquoMicrop-ositioners for microscopy applications based on the stick-slip effectrdquo in Proceedings of the International Symposium onMicromechatronics and Human Science pp 213ndash216 October2000

[8] J M Breguet and R Clavel ldquoStick and slip actuators designcontrol performances and applicationsrdquo in Proceedings of the9th International Symposium onMicromechatronics and HumanScience (MHS rsquo98) pp 89ndash95 November 1998

[9] R Yoshida Y Okamoto and H Okada ldquoDevelopment ofsmooth impact drive mechanism (2nd report)-optimization ofwaveform of driving voltagerdquo Journal of the Japan Society forPrecision Engineering vol 68 no 4 p 536 2002

[10] S B Choi S R Hong and Y M Han ldquoDynamic characteristicsof inertial actuator featuring piezoelectric materials experi-mental verificationrdquo Journal of Sound and Vibration vol 302no 4-5 pp 1048ndash1056 2007

[11] A M Lipanov P V Gulyaev E Y Shelkovnikov and A V Tyu-rikov ldquoAn inertial piezoelectric step drive for subnanosize-accuracy movementsrdquo Instruments and Experimental Tech-niques vol 52 no 5 pp 725ndash726 2009

[12] P V Gulyaev Y K Shelkovnikov A V Tyurikov and N IOsipov ldquoHigh-accuracy inertial rotation-linear piezoelectricdriverdquoRussian Electrical Engineering vol 81 no 10 pp 521ndash5232010

[13] D Mazeika and P Vasiljev ldquoLinear inertial piezoelectric motorwith bimorph discrdquo Mechanical Systems and Signal Processingvol 36 no 1 pp 110ndash117 2013

[14] L Xu and J Xing ldquoAn inertial piezoelectric rotary motorrdquo Pro-ceedings of the Institution of Mechanical Engineers C vol 224no 6 pp 1165ndash1171 2010

[15] C ZhaoUltrasonic Moters Technologies and Applications Scie-nce Press Beijing China 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



middotHousing

Adapting piece

Bolt

Wire

NeedlePiezoelectricityceramic piece

Signalsource

Vibrator

Shaft

Base

(a) Motor structure

BoltNeedle

Vibrator

Base

BearingShaft

Adapting pieceHousing

(b) 3D model

Pzt minus

minus

+

+

(c) Rotor structure and piezoceramic bimorph arrangement

Figure 1 Motor and rotor structures

an inner ring and two ribs connecting the outer ring tothe inner ring The inner ring is mounted on a supportingbearing and the outer ring is used as the inertial mass Thepiezoceramic bimorph is adhered on each side surface of thetwo ribs As soon as a rapid rise input voltage is suppliedto the motor it excites the transverse bending vibration ofthe two ribs Thus the inertial force within the outer ringoccurs which causes inertial torque to be applied to the rotorand makes it rotate Then a slow decreasing input voltage issupplied to the motor so the inertial forces within the outerring are so small that the inertial torque can be balancedby the friction torque between the rotor and the bearingThus the rotor does not rotate The rapidly increasing andslowly decreasing input voltage with a special frequency issupplied to the motor periodically and then the rotor canrotate continuously

For the motor motion a saw-tooth-type electric signal isapplied (see Figure 2) Here 119905 is the time 119879 is the period ofthe saw-tooth cycle 120583 (0 lt 120583 lt 1) is the ratio of the rise timeto the period and 119860 is the amplitude of voltage signal 119881(119905)The voltage signal can be written by

119881 (119905) =

119860

120583119879119905 (0 le 119905 lt 120583119879)

119860

(120583 minus 1) 119879119905 +

119860

1 minus 120583(120583119879 le 119905 le 119879)

(1)

The voltage signal can be written in the Fourier series as

119881 (119905) = 1198860+

infin

sum

119899=1

[119886119899cos (119899120596119905) + 119887

119899sin (119899120596119905)] (2)

where

1198860=

1

119879int

119879

0

119881 (119905) 119889119905

=1

119879(int

120583119879

0

119860

120583119879119905119889119905 + int

119879

120583119879

(119860

(120583 minus 1) 119879119905 +

119860

1 minus 120583)119889119905)

=1

2119860

119886119899=

2

119879int

119879

0

119881 (119905) cos (119899120596119905) 119889119905 =119860 (minus1 + cos (2119899120583120587))

211989921205872120583 (1 minus 120583)

119887119899=

2

119879int

119879

0

119881 (119905) sin (119899120596119905) 119889119905 =119860 sin (2119899120583120587)

211989921205872120583 (1 minus 120583)

(3)

3 The Impulsive Moment

The rotor of the motor can be considered as a beam withinertial mass at its two ends Four piezoceramic bimorphs are


V(t) A

T120583T

t

0


y

x mm

Pzt

ll



1205761=

ℎ

2

1205972119910 (119909 119905)

1205971199092

(4)




1205901 (119909 119905) = minus119890

311198643+

ℎ

2119888119864

11

1205972119910 (119909 119905)

1205971199092

(5)


bimorph 1198643







119901in


119881119901=

119887

2int

119897119901

0

1205901 (119909 119905) 1205761119889119909

=119887ℎ

4

infin

sum

119894=1

119902119894(119905) int

119897119901

0

minus11989031119864312060110158401015840(119909) 119889119909 +

1

2

infin

sum

119894=1

119896119901

119894119895119902119894(119905) 119902119895(119905)

(6)

where 119910(119909 119905) = suminfin

119899=1120601119899(119909)119902119899(119905) 119896119901119894119895

= 119896119901

119895119894= int119897119901

0(119887ℎ24)119888119864

1112060110158401015840

119894

(119909)12060110158401015840

119895(119909)119889119909 120601


119901and 119887 are the



119881119871119875

=1

2int

119897

0

119864119868[1205972119910 (119909 119905)

1205971199092

]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119896119897

119894119895119902119894 (119905) 119902119895 (119905)

(7)

where 119896119897

119894119895= 119896119897

119895119894= int119897

minus11989711986411986812060110158401015840

119894(119909)12060110158401015840




119881 = 4119881119901+ 119881119871119875

(8)


119864119898

=1

2int

119897

minus119897

120588119878[120597119910

120597119905(119909 119905)]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119898119894119895

119902119894(119905) 119902119895(119905)

(9)

where119898119894119895= 119898119895119894

= int119897

minus119897120588119878120601119894(119909)120601119895(119909)119889119909 +119898120601

119894(minus119897)120601119895(minus119897) +119898120601

119894



M 119902 (119905) + C 119902 (119905) + K = F (119905) (10)


119894119895] is the stiffn-

ess matrix 119896119894119895

= 119896119901

119894119895+ 119896119897



119894119895= 119888119895119894

= int119897

0119862119889120601119894




Mn 119902 (119905) + Cn 119902 (119905) + Kn119902 (119905) = Fn (119905) (11)





119865119894 (119905) =

119887ℎ

4(2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119886) 12060110158401015840

119894(119909) 119889119909

+2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119887) 12060110158401015840

119894(119909) 119889119909)

=119887ℎ11989031119881

2ℎ119901

(12060110158401015840

119894(119909119886) + 12060110158401015840

119894(119909119887))

(12)



119902119895=

1

119872119895120596119903119895

int

119905

0

119865119895(120591) 119890minus120585119895120596119895(119905minus120591) sin120596

119903119895(119905 minus 120591) 119889120591 (13)


times10minus6

times10minus6

2

1

0

minus10 05 1 15

0 05 1 15

times10minus3

times10minus3

t (s)

t (s)

y(m

)T

(Nm

) 5

0

minus5

(a) 120596 = 10000 rads

t (s)

t (s)

y(m

)T

(Nm

)

times10minus7

times10minus7

times10minus4

times10minus4

5

0

minus5

15

10

5

0

0 2 4 6 8

0 2 4 6 8

(b) 120596 = 20000 rads



119902119895=

119865119899

1198721198951205962

119895

1198860+

infin

sum

119899=1

1


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895) + 119887119899sin (119899120596119905 minus 120593

119899119895)]

(14)

Here 120593119899119895

= arctan(2120585119895120574119899119895(1 minus 120574

2

119899119895))


119910 (119909 119905) =

infin

sum

119895=1

120601119895 (119909) 119902119895 (119905)

=

infin

sum

119895=1

119865119899120601119895 (119909)

1198721198951205962

119895

times

1198860+

infin

sum

119899=1

radic1198862

119899+ 1198872

119899


119899119895)2

+ (2120585119895120574119899119895)2

times cos(119899120596119905 minus 120593119899119895


119886119899

))

(15)


0 05 1 15 25 3 350

1

2

3

4

2

0

5

10

15



minus5

times104

y(m

)

(rads)

T(N

m)




119879 (119905) =

infin

sum

119895=1

119869ℎ119865119899120601119895(119897)

1198971198721198951205962

119895

infin

sum

119899=1

11989921205962


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895)

+119887119899sin (119899120596119905 minus 120593

119899119895)]

(16)




times10minus6

times104(rads)

T(N

m)

minus05

minus1

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

05

1

15

(a) Mode 1

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(b) Mode 2

times10minus6

minus5

times104(rads)

T(N

m)

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

5

10

(c) Mode 3

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(d) Mode 4






1198771(mm) 119877

2(mm) ℎ (mm) 119897 (mm) 119864 (Gpa)

12 13 1 12 120119897119901(mm) 119887 (mm) ℎ

119901(mm) 119890


10 5 05 minus0046 89 times 103




2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)












5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










V(t) A

T120583T

t

0


y

x mm

Pzt

ll



1205761=

ℎ

2

1205972119910 (119909 119905)

1205971199092

(4)




1205901 (119909 119905) = minus119890

311198643+

ℎ

2119888119864

11

1205972119910 (119909 119905)

1205971199092

(5)


bimorph 1198643







119901in


119881119901=

119887

2int

119897119901

0

1205901 (119909 119905) 1205761119889119909

=119887ℎ

4

infin

sum

119894=1

119902119894(119905) int

119897119901

0

minus11989031119864312060110158401015840(119909) 119889119909 +

1

2

infin

sum

119894=1

119896119901

119894119895119902119894(119905) 119902119895(119905)

(6)

where 119910(119909 119905) = suminfin

119899=1120601119899(119909)119902119899(119905) 119896119901119894119895

= 119896119901

119895119894= int119897119901

0(119887ℎ24)119888119864

1112060110158401015840

119894

(119909)12060110158401015840

119895(119909)119889119909 120601


119901and 119887 are the



119881119871119875

=1

2int

119897

0

119864119868[1205972119910 (119909 119905)

1205971199092

]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119896119897

119894119895119902119894 (119905) 119902119895 (119905)

(7)

where 119896119897

119894119895= 119896119897

119895119894= int119897

minus11989711986411986812060110158401015840

119894(119909)12060110158401015840




119881 = 4119881119901+ 119881119871119875

(8)


119864119898

=1

2int

119897

minus119897

120588119878[120597119910

120597119905(119909 119905)]

2

119889119909 =1

2

infin

sum

119894=1

infin

sum

119895=1

119898119894119895

119902119894(119905) 119902119895(119905)

(9)

where119898119894119895= 119898119895119894

= int119897

minus119897120588119878120601119894(119909)120601119895(119909)119889119909 +119898120601

119894(minus119897)120601119895(minus119897) +119898120601

119894



M 119902 (119905) + C 119902 (119905) + K = F (119905) (10)


119894119895] is the stiffn-

ess matrix 119896119894119895

= 119896119901

119894119895+ 119896119897



119894119895= 119888119895119894

= int119897

0119862119889120601119894




Mn 119902 (119905) + Cn 119902 (119905) + Kn119902 (119905) = Fn (119905) (11)





119865119894 (119905) =

119887ℎ

4(2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119886) 12060110158401015840

119894(119909) 119889119909

+2int

119897

minus119897

119890311198643120575 (119909 minus 119909

119887) 12060110158401015840

119894(119909) 119889119909)

=119887ℎ11989031119881

2ℎ119901

(12060110158401015840

119894(119909119886) + 12060110158401015840

119894(119909119887))

(12)



119902119895=

1

119872119895120596119903119895

int

119905

0

119865119895(120591) 119890minus120585119895120596119895(119905minus120591) sin120596

119903119895(119905 minus 120591) 119889120591 (13)


times10minus6

times10minus6

2

1

0

minus10 05 1 15

0 05 1 15

times10minus3

times10minus3

t (s)

t (s)

y(m

)T

(Nm

) 5

0

minus5

(a) 120596 = 10000 rads

t (s)

t (s)

y(m

)T

(Nm

)

times10minus7

times10minus7

times10minus4

times10minus4

5

0

minus5

15

10

5

0

0 2 4 6 8

0 2 4 6 8

(b) 120596 = 20000 rads



119902119895=

119865119899

1198721198951205962

119895

1198860+

infin

sum

119899=1

1


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895) + 119887119899sin (119899120596119905 minus 120593

119899119895)]

(14)

Here 120593119899119895

= arctan(2120585119895120574119899119895(1 minus 120574

2

119899119895))


119910 (119909 119905) =

infin

sum

119895=1

120601119895 (119909) 119902119895 (119905)

=

infin

sum

119895=1

119865119899120601119895 (119909)

1198721198951205962

119895

times

1198860+

infin

sum

119899=1

radic1198862

119899+ 1198872

119899


119899119895)2

+ (2120585119895120574119899119895)2

times cos(119899120596119905 minus 120593119899119895


119886119899

))

(15)


0 05 1 15 25 3 350

1

2

3

4

2

0

5

10

15



minus5

times104

y(m

)

(rads)

T(N

m)




119879 (119905) =

infin

sum

119895=1

119869ℎ119865119899120601119895(119897)

1198971198721198951205962

119895

infin

sum

119899=1

11989921205962


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895)

+119887119899sin (119899120596119905 minus 120593

119899119895)]

(16)




times10minus6

times104(rads)

T(N

m)

minus05

minus1

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

05

1

15

(a) Mode 1

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(b) Mode 2

times10minus6

minus5

times104(rads)

T(N

m)

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

5

10

(c) Mode 3

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(d) Mode 4






1198771(mm) 119877

2(mm) ℎ (mm) 119897 (mm) 119864 (Gpa)

12 13 1 12 120119897119901(mm) 119887 (mm) ℎ

119901(mm) 119890


10 5 05 minus0046 89 times 103




2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)












5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus6

times10minus6

2

1

0

minus10 05 1 15

0 05 1 15

times10minus3

times10minus3

t (s)

t (s)

y(m

)T

(Nm

) 5

0

minus5

(a) 120596 = 10000 rads

t (s)

t (s)

y(m

)T

(Nm

)

times10minus7

times10minus7

times10minus4

times10minus4

5

0

minus5

15

10

5

0

0 2 4 6 8

0 2 4 6 8

(b) 120596 = 20000 rads



119902119895=

119865119899

1198721198951205962

119895

1198860+

infin

sum

119899=1

1


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895) + 119887119899sin (119899120596119905 minus 120593

119899119895)]

(14)

Here 120593119899119895

= arctan(2120585119895120574119899119895(1 minus 120574

2

119899119895))


119910 (119909 119905) =

infin

sum

119895=1

120601119895 (119909) 119902119895 (119905)

=

infin

sum

119895=1

119865119899120601119895 (119909)

1198721198951205962

119895

times

1198860+

infin

sum

119899=1

radic1198862

119899+ 1198872

119899


119899119895)2

+ (2120585119895120574119899119895)2

times cos(119899120596119905 minus 120593119899119895


119886119899

))

(15)


0 05 1 15 25 3 350

1

2

3

4

2

0

5

10

15



minus5

times104

y(m

)

(rads)

T(N

m)




119879 (119905) =

infin

sum

119895=1

119869ℎ119865119899120601119895(119897)

1198971198721198951205962

119895

infin

sum

119899=1

11989921205962


119899119895)2

+ (2120585119895120574119899119895)2

times [119886119899cos (119899120596119905 minus 120593

119899119895)

+119887119899sin (119899120596119905 minus 120593

119899119895)]

(16)




times10minus6

times104(rads)

T(N

m)

minus05

minus1

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

05

1

15

(a) Mode 1

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(b) Mode 2

times10minus6

minus5

times104(rads)

T(N

m)

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

5

10

(c) Mode 3

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(d) Mode 4






1198771(mm) 119877

2(mm) ℎ (mm) 119897 (mm) 119864 (Gpa)

12 13 1 12 120119897119901(mm) 119887 (mm) ℎ

119901(mm) 119890


10 5 05 minus0046 89 times 103




2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)












5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus6

times104(rads)

T(N

m)

minus05

minus1

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

05

1

15

(a) Mode 1

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(b) Mode 2

times10minus6

minus5

times104(rads)

T(N

m)

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

5

10

(c) Mode 3

times10minus6

times104(rads)

T(N

m)

minus2

minus4

120583 = 07

120583 = 08

120583 = 09

2 4 6 8 10

0

2

4

6

(d) Mode 4






1198771(mm) 119877

2(mm) ℎ (mm) 119897 (mm) 119864 (Gpa)

12 13 1 12 120119897119901(mm) 119887 (mm) ℎ

119901(mm) 119890


10 5 05 minus0046 89 times 103




2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)












5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 4 6 8 10

0

2

4

6

8

10

12

times10minus5

times104

Mode 3

(rads)

T(N

m)

minus2

120585 = 001120585 = 008120585 = 02

(a) 120585 change

2 4 6 8 10

0

5

10

15

20

25times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

A = 30A = 40A = 50

(b) 119860 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)

e31 = 0046e31 = 0015e31 = 0024

(c) 11989031 change

2 4 6 8 10

0

5

10

15times10minus6

minus5

times104

Mode 3

(rads)

T(N

m)












5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 10 15

0

5

10

15

20


minus5

(rads)

T(N

m)

times104

times10minus4











Sizenumber

1198771

(mm)1198772

(mm)119896

(mm)119897

(mm)ℎ

(mm)120596

(rads)1 13 12 5 12 1 842542 18 15 10 15 2 1617683 25 175 15 175 3 49336







should beselected




should be taken




6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6200 6250 6300 63500

10

20

30

40

50

Frequence (Hz)

Spee

d (r

min

)

(a) 120596 changes 119881 = 100V

40 60 80 100 120 140 1600

20

40

60

80

100

120

Spee

d (r

min

)

Voltage (V)

(b) 120596 = 6310Hz 119881 changes

Frequence (Hz)6280 6290 6300 6310 6320 6330

01

015

02

025

03

035

Torq

ue (N

mm

)

(c) 120596 changes 119881 = 100V

Torq

ue (N

mm

)

40 60 80 100 120 140 160

01

02

03

04

05

Voltage (V)

(d) 120596 = 6310Hz 119881 changes



Signalvoltage (V)



Measuredtorque(Nmm)

Relativeerrors ()

75 098 037 024 35100 131 050 035 30125 17 062 039 37150 201 075 044 41175 23 087 052 40









5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











5 Conclusions





Acknowledgment


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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