Research Article A Broadband Vibration-Based Energy ...downloads.hindawi.com/journals/sv/2016/9614842.pdf · a broadband vibration-based energy harvester. e equations for the voltage

Research ArticleA Broadband Vibration-Based Energy Harvester Using an Arrayof Piezoelectric Beams Connected by Springs

V Meruane and K Pichara

Department of Mechanical Engineering Universidad de Chile Beauchef 851 9370456 Santiago Chile

Correspondence should be addressed to V Meruane vmeruaneinguchilecl

Received 2 June 2015 Revised 25 August 2015 Accepted 27 August 2015

Academic Editor Carlo Trigona

Copyright copy 2016 V Meruane and K Pichara This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Piezoelectric cantilevered beams have been widely used as vibration-based energy harvesters Nevertheless these devices have anarrow frequency band and if the excitation is slightly different there is a significant drop in the level of power generated To handlethis problem the present investigation proposes the use of an array of piezoelectric cantilevered beams connected by springs asa broadband vibration-based energy harvester The equations for the voltage and power output of the system are derived basedon the analytical solution of the piezoelectric cantilevered energy harvester with Euler-Bernoulli beam assumptions To study theadvantages and disadvantages of the proposed system the results are compared with those of an array of disconnected beams (withno springs) The analytical model is validated with experimental measurements of three bimorph beams with and without springsThe results show that connecting the array of beams with springs allows increasing the frequency band of operation and increasingthe amount of power generated

1 Introduction

The idea of harvesting energy from the environment result-ing in self-powered monitoring systems is fascinating andhas attracted the attention of many researchers This isdemonstrated by the large numbers of investigations in thearea [1ndash8] Examples of remote sensors that could benefitfrom energy harvesting are weather monitoring sea surfacewater monitoring tactical military surveillance monitoringof equipment and structures oceanography and volcanoand seismic monitoring to name a few For most of theseapplications renewable power sources are needed for long-term remotemonitoring Some of the different power sourcesavailable are thermoelectric energy mechanical vibrationshydroelectric power wind energy solar power and ambientradio frequency

Mechanical vibrations are an attractive source due to theirhigh availability in certain environments The transductionmechanisms that can be used to convert ambient vibrationinto electrical energy are electromagnetic electrostatic andpiezoelectric Among these mechanisms piezoelectric trans-duction has received great attention in the last years due

to its ability to directly convert strain energy into usableelectric energy [1 9] In piezoelectric energy harvesting themechanical energy of vibrations is transformed by a piezo-electric material to electrical energy When a piezoelectricmaterial is deformed the central molecules in the crystalbecome polarized and form a dipole If the dipoles arearranged then two surfaces of thematerial become positivelyand negatively charged This property can be exploited totransform mechanical strain into electrical energy Figure 1presents a scheme of two cantilever piezoelectric powergenerators The generators are submitted to base excitationwhich causes the beams to vibrate inducing an alternatingvoltage The voltage generated by the piezoelectric elementscan be used to power a device directly or to charge arechargeable battery In the case of a unimorph beam alayer of piezoceramic material is attached to a metal layer(substructure) whereas in a bimorph configuration a metallayer is sandwiched between two piezoelectric layers In bothcases a proofmass is attached at the end to increase the strain

The main problem of conventional energy harvestersis their narrow frequency band of operation Conventionalenergy harvesters as the ones shown in Figure 1 have

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9614842 13 pageshttpdxdoiorg10115520169614842

2 Shock and Vibration

Mt Rl (t)

+

minus

PiezoceramicSubstructure

(a)

Mt Rl (t)

+

minus


(b)

Figure 1 Cantilever piezoelectric energy configurations (a) unimorph (b) bimorph (parallel connection)

an operating frequency equal to their resonance frequencyand if the excitation is slightly different there is a significantdrop in the level of power generated This problem has ledto the search for broadband energy harvesters Twiefel andWestermann [10] present a survey of broadband techniquesfor vibration-based energy harvesting These techniques areclassified as follows

(i) Linear generators arrays combined modes

(ii) Nonlinear generators bimultiple stable vibroimpactresonant tuning

(iii) Advanced electronic networks switching networkssynthetic impedance

According to the researchers there is no universal solutiondepending on the application one technique may be betterthan the others For example linear generators are moreefficient for stochastic excitations whereas nonlinear tech-niques are more suitable for harmonic excitations or very lowexcitation frequencies

In the case of linear generators recent research hasbeen focused on developing systems that exhibit multipleresonances that are closely spaced The simplest solution isto use multiple cantilevers The cantilevers are built withdifferent geometries (large tip mass or width) so each beamhas a slightly different resonant frequency thus increasing thebandwidth of the full system A first design of a cantileverarray was presented by Shahruz [11 12] He concluded thatif the dimensions of the beams and masses are chosenappropriately then the system performs as a broadbandharvester However the maximal frequency band is limitedand is independent of the beams dimensions This approachhas been studied extensively since then [13ndash15]

Another method is to use structures with two or moredegrees of freedom resulting in two or more resonantfrequencies that are closely spaced [16ndash19] Z Yang andJ Yang [18] analyzed the power generated by the flexuralvibrations of two piezoelectric beams connected by springsthey showed that this structure can be used as a broadbandenergy harvester Nevertheless the model presented by theauthors is restricted to two beams and the extension tomore than two beams is not trivial Springs have also beenused in MEMS-based energy harvesters where it has been

demonstrated that the use of nonlinear springs increases thegenerated power [20ndash22]

The research community related to piezoelectric energyharvesting is very wide and includes researchers frommechanical electrical materials and civil engineering Thishas led to the development of many different models for thesame problem a piezoelectric cantilevered beam Most ofthese models use simplifications such as oversimplifying ofthe piezoelectric coupling as viscous damping [23] using ofthe static deflection of the beam in a dynamic model [24 25]no inclusion of the piezoelectric effect in the mechanicalmodel [26] or approximate solutions as the Rayleigh-Ritzapproximation [27 28] Erturk and Inman [29] providea comprehensive discussion of the different simplificationsfound in literature to model piezoelectric beams highlight-ing the common indiscretions and providing correctionsand necessary clarifications for researchers from differentengineering fields The best solution can be obtained bysolving the continuous electromechanical model analyticallyDifferent approaches for solving the continuousmodel can befound in literature [30ndash33]

In this paper we present a general approach to character-ize multiple piezoelectric beams connected with springs Theanalytical solution of the cantilevered piezoelectric energyharvester with Euler-Bernoulli beam assumptions proposedby Erturk and Inman [33] is extended to the case of an arrayof piezoelectric beams connected by springs The proposedconfiguration is attractive since it combines the properties ofbeam arrays and of systems withmultiple degrees of freedomFurthermore the results demonstrate that connecting anarray of beams with springs allows increasing the frequencyband of operation and increasing the amount of powergenerated when compared with an array of disconnectedbeams

The remainder of this work is organized as followsSection 2 presents the structure under investigation andthe methodology to compute its modes shapes and naturalfrequencies Section 3 provides the derivation of the elec-tromechanical equations of a bimorph cantilever beam InSection 4 the equations of the output voltage and power ofan array of bimorph cantilever beams connected by springsare derived Section 5 presents analytical and experimentalresults for different combinations of tip masses and springsconstants comparing the results with those of an array ofdisconnected beams (with no springs) Finally conclusionsand forthcoming work are presented in Section 6

Shock and Vibration 3

Mt1

Mt2

MtN

k1

k2

kN

(a) Full system

kiminus1

kiwi(x t)

Mtiminus1

Mti

Mti+1

X

(b) Scheme of the 119894th beam

Figure 2 Structure consisting in an array of piezoelectric beams connected by springs

2 Structure and Vibration Modes

The structure under investigation is an array of piezoelectricbeams connected by springs as illustrated in Figure 2(a) Allbeams are clamped at the left-end to a vertical wallThe right-end of each beam is connected to a concentrated tip massand to its neighbors beams by springs The inertia of theconcentrated masses is neglected All beams have identicaldimensions and the only differences are the tip masses

To derive the vibration modes of this system it isnecessary to study the boundary conditions of each beam sep-arately Looking at the 119894th beam as shown in Figure 2(b) andassuming a separation-of-variables solution the transversedisplacement of the 119894th beam can be expressed as 119908

119894(119909 119905) =

120601119894(119909)120578(119905) The function 120601

119894(119909) represents the mode shape and

depends on the boundary conditions In an Euler-Bernoullibeam the general solution for 120601

119894(119909) is

120601119894 (119909) = 120582

119894

1sin (120573119909) + 120582119894

2cos (120573119909) + 120582119894

3sinh (120573119909)

+ 120582119894

4cosh (120573119909)

(1)

The boundary conditions at the left-end are

120601119894 (0) = 0

120597120601119894 (0)

120597119909= 0

(2)

which implies that

120582119894

4= minus120582119894

2

120582119894

3= minus120582119894

1

(3)

Therefore the mode shape can be written as

120601119894 (119909) = 120582

119894

1[sin (120573119909) minus sinh (120573119909)]

+ 120582119894

2[cos (120573119909) minus cosh (120573119909)]

(4)

At the right-end the bending moment and shear forcemust satisfy the following conditions

1198841198681205972119908119894

1205971199092= 0

120597

120597119909[119884119868

1205972119908119894

1205971199092] = minus119872

119905119894

1205972119908119894

1205971199052minus 119896119894(119908119894minus 119908119894+1)

minus 119896119894minus1(119908119894minus 119908119894minus1)

(5)

where 119884 is Youngrsquos modulus 119868 is the area moment of inertiaof the cross section 119872

119905119894is the 119894th tip mass and 119896

119894is the

stiffness of the 119894th spring Replacing that 119908119894(119909 119905) = 120601

119894(119909)120578(119905)

and 1205972120578(119905)1205971199052 = minus1205962119899120578(119905) 120596

119899being the natural frequency

yields

1205972120601119894 (119871)

1205971199092= 0

1198841198681205973120601119894 (119871)

1205971199093= 1205962

119899119872119905119894120601119894 (119871) minus 119896119894 [120601119894 (119871) minus 120601119894+1 (119871)]

minus 119896119894minus1[120601119894 (119871) minus 120601119894minus1 (119871)]

(6)


where 119871 is the length of the beam Replacing the definition of120601119894(119909) from (4) the following equations are obtained

[

0 0 1205721

1205722

0 0

1205723(119896119894minus1) 1205724(119896119894minus1) 1205725(119872119905119894 119896119894 119896119894minus1) 1205726(119872119905119894 119896119894 119896119894minus1) 1205723(119896119894) 1205724(119896119894)]

[[[[[[[[[[[[

[

120582119894minus1

1

120582119894minus1

2

120582119894

1

120582119894

2

120582119894+1

1

120582119894+1

2

]]]]]]]]]]]]

]

=

[[[[[[[[[[[

[

0

0

0

0

0

0

]]]]]]]]]]]

]

(7)

with

1205721= sin (120573119871) + sinh (120573119871)

1205722= cos (120573119871) + cosh (120573119871)

1205723(119896119894) = 119896119894(sinh (120573119871) minus sin (120573119871))

1205724(119896119894) = 119896119894(cosh (120573119871) minus cos (120573119871))

1205725(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (sin (120573119871) minus sinh (120573119871))

+ 1198841198681205733(cos (120573119871) + cosh (120573119871))

1205726(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (cos (120573119871) minus cosh (120573119871))

+ 1198841198681205733(sinh (120573119871) minus sin (120573119871))

(8)

Equation (7) can be obtained for each of the119873 beams inthe structure Particular cases are the first and last beams Inthe case of the first beam there is no previous neighbor thusthe following equation is obtained

[

1205721

1205722

0 0

1205725(1198721199051 1198961 0) 120572

6(1198721199051 1198961 0) 120572

3(1198961) 1205724(1198961)]

[[[[[[

[

1205821

1

1205821

2

1205822

1

1205822

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(9)

whereas for the last beam there is no next neighbor and theresulting equations are

[

0 0 1205721

1205722

1205723(119896119873minus1) 1205724(119896119873minus1) 1205725(119872119905119873 0 119896119873minus1) 1205726(119872119905119873 0 119896119873minus1)]

[[[[[[

[

120582119873minus1

1

120582119873minus1

2

120582119873

1

120582119873

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(10)

Coupling all equations yields a system of 2times119873 equationswhich can be written as the following matrix equation

C120582 = 0 (11)

where C is a coefficient matrix and 120582 = [12058211 1205821

2 1205822

1 1205822

2

120582119873

1 120582119873

2]119879 This equation can have nonzero solution for 120582 only

if the determinant of the coefficient matrix vanishes (ie thecoefficient matrix is singular) Setting the determinant of Cequal to zero yields the characteristic equation that is satisfiedby an infinite number of choices for 120573 denoted by 120573

119903 The

vector of unknowns 120582 is calculated by replacing 120573119903in the

coefficient matrix C and then computing its null space Foreach value of 120573

119903a different vector 120582

119903is obtained

Given 120573119903and 120582

119903 the mode shapes are given by

120601119903119894 (119909) = 120582

119894

1199031[sin (120573

119903119909) minus sinh (120573

119903119909)]

+ 120582119894

1199032[cos (120573

119903119909) minus cosh (120573

119903119909)]

(12)

The undamped natural frequency associated with the 119903thmode is given by

120596119899119903= 1205732

119903radic119884119868

119898 (13)

where119898 is the mass per unit length As an example Figure 3illustrates the vibration modes of three elastically connected


120573rL = 1536

(a) Mode 1

120573rL = 1573

(b) Mode 2

120573rL = 1640

(c) Mode 3

120573rL = 4192

(d) Mode 4 (e) Mode 5 (f) Mode 6

(g) Mode 7 (h) Mode 8 (i) Mode 9

Figure 3 First nine vibration modes of three elastically connected beams with identical tip masses and spring constants (119872119905119894=13

=

03119898119871 11989612= 03119864119868119871

3)

beams In this example all beams have a mass tip of 03119898119871and all springs have a stiffness of 031198841198681198713

3 Electromechanical Model of a BimorphPiezoelectric Beam

31 Mechanical Equation Let us consider the bimorph can-tilever beam shown in Figure 4 The bimorph beam consistsin two layers of piezoelectric material (PZT) bonded to asubstructure layer Assuming an Euler-Bernoulli beam theequation of motion can be written as [33]

1205972119872(119909 119905)

1205971199092+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(14)

b

x(t)

Mt

hp

hp

hs

PZT

PZTSubstructureY0e

i120596t

Figure 4 Scheme of the bimorph piezoelectric beam

where 119908rel(119909 119905) is the transverse deflection of the beamrelative to the base 119908

119887(119905) is the base displacement119872(119909 119905) is

the internal bending moment 119888119904is the equivalent coefficient

of strain rate damping 119888119886is the viscous air damping 119868 is the

area moment of inertia of the cross section119898 is the mass perunit length 119872

119905is the tip mass and 120575(119909) is the Dirac delta

function


The internal moment is given by the integral of the stressdistribution over the cross section of the beam shown in

119872(119909 119905) = minus119887 [int

minusℎ1199042

minusℎ119901minusℎ1199042

119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)

where 119887 is the width of the beam ℎ119901is the thickness of

each PZT layer ℎ119904is the thickness of the substructure (see

Figure 4) and 119879 is the stress Superscripts 119901 and 119904 stand forPZT and substructure respectively The stress distributionacross the section is given by the strain-stress relations of thesubstructure and PZT layers which are given by

119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)

where 119878 is the strain 119884 is Youngrsquos modulus 119889 is thepiezoelectric constant and 119864 is the electric fieldThe bendingstrain can be expressed in terms of the radius of curvature as

119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)

Since the piezoelectric layers are connected in parallelthey have the same voltage but the instantaneous electricfields are in opposite directions The electric field can bewritten in terms of the voltage across the PZT V(119905) and thethickness of the PZT layer ℎ

119901 by the following equation

1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901

in the bottom layer(19)

Replacing (16) to (19) in (15) we obtain

119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)

Equation (20) can be expressed as

119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)

where 119871 is the length of the beam and Φ the Heavisidefunction It has been assumed that the PZT layer covers

the entire length of the beam The constant 119884119868 represents thebending stiffness of the beam and is given by

119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)

and the term 120599 represents the coupling between the strain andthe voltage and is given by

120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)

Substituting (21) in (14) we obtain the mechanical equa-tion of motion with electrical coupling

1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)

32 Electrical Equation To derive the electrical equation letus start with the piezoelectric constitutive relation given in

119863 = 11988931119879119901+ 120576119879

33119864 (25)

where119863 is the electric displacement and 12057611987933is the permittivity

at constant stress Replacing the definition of 119879119901 from (17)yields

119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879

33 The strain 119878119901 evaluated at the middle

of the PZT layer is given by

119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ

1199012 is the distance of the center of the

PZT layer to the neural axis Thus

119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)

The electric charge is obtained by integrating the electricdisplacement over the area of the beamThe resulting expres-sion is

119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)


Therefore the current generated by one PZT layer is

119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)

Since both layers are connected in parallel the currentgenerated by the bimorph beam is given by

119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)

If the beam is connected directly to a resistive load the voltageis given by V(119905) = 119877

119897119894(119905) Substituting this relation in (31) yields

V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)

The electrical equation with mechanical coupling isobtained by rearranging (32)

V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)

4 Voltage and Power Output of an Array ofPiezoelectric Beams Connected by Springs

Equations (24) and (33) are the electromechanical equationsfor a single bimorph beam under transverse vibrations Letus consider the case of multiple beams connected by springswith a parallel electrical connection as shown in Figure 5 Inthat case the mechanical equation of motion is given by

1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl

Figure 5 A parallel electrical connection of an array of bimorphbeams connected by springs

where V119894(119905) is the voltage of the 119894th beam Since the beams are

electrically connected in parallel they have the same voltagethus V

1(119905) = V

2(119905) = sdot sdot sdot = V

119899(119905) = V(119905) It has been assumed

that all the beams are identical except for the tip mass Itshould be noted that the springs are included as boundaryconditionsTherefore they affect the vibrationmodes but notthe equation of motion

The solution for the relative motion of the system can berepresented by a sum of its mode shapes as

119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578

119903(119905) are the 119903th mode shape and modal

coordinate respectively The 119903th mode shape can be definedby segments as

120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871

120601119903119873 (119909 minus (119873 minus 1) 119871) (119873 minus 1) 119871 le 119909 le 119873119871

(36)

where 120601119903119894

is the 119903th mode shape of the 119894th beam 119871 is thelength of one beam and119873 is the number of beams

Replacing (35) themechanical equation ofmotion can bewritten as

[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)

Assuming a harmonic basemotion119908119887(119909 119905) = 119884

0119890119895120596119905 and

a harmonic response 120578119903(119905) = 119873

119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)

Rearranging (39) the amplitude of the temporal term119873119903

can be written as follows

119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)

For simplicity (40) can be expressed as

119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)

The electrical equation needs to be derived for each beamindividually The relative motion for the 119894th beam is given by

119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)

Replacing (43) in (30) yields

119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)

If the beams are connected directly to a resistive load thevoltage is given by V(119905) = 119877

119897119894(119905) = 119877

119897sum119894119894119894(119905) substituting this

condition equation (44) becomes

V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)

Replacing the harmonic response condition in (45) yields

119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)

Rearranging (46) we obtain

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)

Replacing 119873119903from (41) in (47) the following equation is

obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)

Rearranging (49) the voltage amplitude can be written asshown in

119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)

The voltage FRF is defined as the ratio of the voltageoutput to the base acceleration119881minus1205962119884

0 The instantaneous

power output can be obtained from the relation 119875(119905) =V2(119905)119877

119897Therefore the FRF of the power output is simply the

square of the voltage FRF divided by 119877119897

5 Results

51 Analytical This section presents the results of the powerFRF for different number of beams and different combina-tions of tip masses and spring constants in the array Thegeometric material and electromechanical parameters of thebeams are given in Table 1 These are the parameters of theexperimental beams used in the next section

The voltage is computed by (50) and the power FRF iscomputed as 1198812[119877

119897(12059621198840)2] The value of the external load


One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected

Figure 6 Power FRF with an increasing number of identical bimorph beams119872119905119894=16

= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected

Figure 7 Power FRF with an increasing number of bimorph beams with different tip masses Two beams1198721199051= 03119898119871119872

1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3

Table 1 Parameters of the bimorph beams

Length of the beam 119871 (mm) 482Width of the beam 119887 (mm) 10Thickness of the substructure ℎ

119904(mm) 019

Thickness of the PZT ℎ119901(mm) 024

Youngrsquos modulus of the substructure 119884119904(GPa) 404

Youngrsquos modulus of the PZT 119884119901(GPa) 55

Mass density of the substructure 120588119904(kgm3) 1519

Mass density of the PZT 120588119901(kgm3) 8140

Piezoelectric constant 11988931(pmV) minus180

Permittivity 12057611990433(nFm) 1328

Strain rate damping 119888119904119868 (kgm4s) 116 times 10minus6

Viscous air damping 119888119886(kgs) 04

resistance is defined for each case as the optimum resistorthat maximizes the power output To study the advantagesand disadvantages of the proposed system the results arecompared with those obtained by an array of disconnectedbeams (with no springs)

First the case of identical beams is investigated theresults are shown in Figure 6(a) It is obtained that in thecase of identical beams the only modes that are excitedare the modes where all beams move in phase such asmodes 1 4 and 7 in Figure 3 In this case it is also foundthat the natural frequencies do not depend on the springstiffness because all beams are in phase and the springs donot deform Furthermore the results are exactly the same asthe ones obtained with disconnected beams as illustrated inFigure 6(b)

In the case of beams with different tip masses the resultsof the FRF power do depend on the springs connecting thebeams as shown in Figure 7 If the beams are disconnectedand new beams with different tip masses are added to thearray then the frequency band of the device is increasedobtaining a similar power amplitude for all resonant frequen-cies On the other hand if the beams are connected by springand new beams are included in the array then the frequencyband is increased but the amplitude related to each resonantfrequency varies

For an array of piezoelectric beams connected by springsthe frequency band of operation and the power amplitude


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL

(a) Varying tip masses (119896 = 031198841198681198713)

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k = 02YIL3k = 03YIL3

k = 04YIL3

(b) Varying spring constants (1198721199051 = 03119898119871 1198721199052 = 04119898119871)

Figure 8 Power FRF of two bimorph beams with different combinations of tip masses and spring constants

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

02mL 03mL 04mL02mL 02mL 04mL

03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =

k =k = 02YIL3 02YIL302YIL3 04YIL3

02YIL3 06YIL3

(b) Varying spring constants (1198721199051 = 02119898119871 1198721199052 = 03119898119871 1198721199053 =04119898119871)

Figure 9 Power FRF of three bimorph beams with different combinations of tip masses and spring constants

associated with each resonant frequency depend on thecombination of tipmasses and springs constantsThis ismoreevident in Figures 8 and 9 where different combinations ofsprings and tip masses are studied for the case of two andthree beams In general increasing the tip mass of the beamsdecreases someof the resonant frequencies and increasing thespring stiffness increases the frequency band

To compare the performances of an array of beamsconnected by springs and one of the disconnected beams acase of ten beams with both situations is plotted in Figure 10It is possible to see that the array with connected beamshas a slightly wider frequency band which increases withan increment of the springs constants Furthermore bycomparing the areas below both curves in the frequencyrange 50ndash200Hz the array with connected beams providesan area between 20 and 36 larger than the array withdisconnected beams Thus by connecting the beams theperformance of the array can be improved It should benoted that the previous results are obtained with arbitraryvalues for the tip masses and spring constantsTherefore thisperformance can be further improved by a proper design

52 Experimental To validate the proposed model a pre-liminary experimental campaign was performed Figure 11presents the experimental setup It consists of a signal genera-tor connected to a power amplifier and to an electrodynamicshaker which produces a controlled base vibration An arrayof three piezoelectric bimorph beams is mounted over theshaker Figure 12 shows this array The cantilever bimorphbeams have a tip mass at the free-end and can be connectedwith springs between them Three experiments were per-formed In the first case no springs were used and eachbeam was connected independently to a 1 kΩ resistive loadin this case the voltage generated by each beam is monitoredindividually In the second case no springs were used andthe beams were electrically connected in parallel to a resistiveload of 1 kΩ In the last case the beams were connected withsprings between them and they were electrically connected inparallel to a resistive load of 1 kΩ

The properties of the bimorph beams are the ones givenin Table 1 the tip masses are119872

1199051= 65119898119871119872

1199052= 35119898119871

1198721199053= 19119898119871 and the spring constants are 119896

1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713

Figure 10 Power FRF with ten bimorph beams1198721199051= 01119898119871119872

1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871

Figure 11 Experimental setup

Figure 12 Experimental bimorph beams

During the three experiments the system is subjected toa harmonic base displacement with frequencies in the range20ndash70Hz The results for the first experiment are presentedin Figure 13(a) and are compared with the ones obtainedby an analytical model in Figure 13(b) Both analytical andexperimental curves are very similar The only difference isthat the experimental power increases from beams 1 to 3 andthe analytical model predicted the opposite behavior Thisunexpected behavior seems to be related to a resonance ofthe supporting structure in the three experimental curves itis possible to see an increment of the power near 60Hz whichcan be explained by a resonance of the structure

Figure 14 presents the experimental and analytical resultsfor the second and third experiments It is possible to seea good agreement between the experimental and analyticalcurves although similarly to the results in Figure 13(a) thereis an increment of the power near 60Hz that confirms astructural resonance

Comparing the areas below both experimental curves inFigure 13(b) in the frequency range 20ndash70Hz the array withconnected beams provides an area 5 larger than the arraywith disconnected beamswhich confirms that the addition ofsprings can increment the generated powerThese results canbe improved by using softer springs which have lower energylosses and by solving the structural resonance problem


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical

Figure 13 Power FRF for the three beams with no springs and independent electrical connections to 1 kΩ resistive loads (a) experimentalmeasurement (b) analytical results

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

ConnectedDisconnected

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical

Figure 14 Power FRF for the three beams with and without springs and electrical connection in parallel to a resistive load of 1 kΩ(a) experimental measurement (b) analytical results

6 Conclusions

This paper investigated the use of an array of bimorphbeams connected by springs as a broadband vibration-basedenergy harvester The equations for the voltage and poweroutput of the full system have been deduced based on theanalytical solution of the cantilevered piezoelectric energyharvester with Euler-Bernoulli beam assumptions To studythe advantages and disadvantages of the proposed system theresults are compared with those of an array of disconnectedbeams (with no springs) The analytical model is validatedwith experimental measurements of three bimorph beamswith and without springs

Both analytical and experimental results show that con-necting the array of beams with springs allows increasingthe frequency band of operation and increasing the amountof power generated Thus the maximal frequency band fora certain array of beams can be extended by connectingthem by springs and at the same time the output power isincreased The performance of this system can be optimizedfor specific applications by a proper selection of the tipmassesand spring constants

The preliminary experimental results are consistent withthe analytical ones but further research is necessary toimprove the experimental setup and to investigate differentexperimental conditions such as variations of the springsstiffness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S R Anton and H A Sodano ldquoA review of power harvestingusing piezoelectric materials (2003ndash2006)rdquo Smart Materialsand Structures vol 16 no 3 article R1 2007

[2] S Chalasani and J M Conrad ldquoA survey of energy harvestingsources for embedded systemsrdquo in Proceedings of the IEEESoutheastCon pp 442ndash447 April 2008

[3] K A Cook-Chennault NThambi and AM Sastry ldquoPoweringMEMS portable devicesmdasha review of non-regenerative andregenerative power supply systems with special emphasis on


piezoelectric energy harvesting systemsrdquo Smart Materials andStructures vol 17 no 4 Article ID 043001 2008

[4] A Dewan S U Ay M N Karim and H Beyenal ldquoAlternativepower sources for remote sensors a reviewrdquo Journal of PowerSources vol 245 pp 129ndash143 2014

[5] A Harb ldquoEnergy harvesting state-of-the-artrdquo RenewableEnergy vol 36 no 10 pp 2641ndash2654 2011

[6] S Saadon and O Sidek ldquoA review of vibration-based MEMSpiezoelectric energy harvestersrdquo Energy Conversion and Man-agement vol 52 no 1 pp 500ndash504 2011

[7] H A Sodano D J Inman and G Park ldquoA review of powerharvesting from vibration using piezoelectric materialsrdquo Shockand Vibration Digest vol 36 no 3 pp 197ndash205 2004

[8] R J M Vullers R van Schaijk I Doms C Van Hoofand R Mertens ldquoMicropower energy harvestingrdquo Solid-StateElectronics vol 53 no 7 pp 684ndash693 2009

[9] R Calio U B Rongala D Camboni et al ldquoPiezoelectric energyharvesting solutionsrdquo Sensors vol 14 no 3 pp 4755ndash4790 2014

[10] J Twiefel and H Westermann ldquoSurvey on broadband tech-niques for vibration energy harvestingrdquo Journal of IntelligentMaterial Systems and Structures vol 24 no 11 pp 1291ndash13022013

[11] S M Shahruz ldquoDesign of mechanical band-pass filters forenergy scavengingrdquo Journal of Sound and Vibration vol 292no 3ndash5 pp 987ndash998 2006

[12] S M Shahruz ldquoLimits of performance of mechanical band-pass filters used in energy scavengingrdquo Journal of Sound andVibration vol 293 no 1-2 pp 449ndash461 2006

[13] S M Shahruz ldquoDesign of mechanical band-pass filters forenergy scavenging multi-degree-of-freedom modelsrdquo Journalof Vibration and Control vol 14 no 5 pp 753ndash768 2008

[14] J-Q Liu H-B Fang Z-Y Xu et al ldquoA MEMS-based piezo-electric power generator array for vibration energy harvestingrdquoMicroelectronics Journal vol 39 no 5 pp 802ndash806 2008

[15] HC Lin PHWu I C Lien andYC Shu ldquoAnalysis of an arrayof piezo-electric energy harvesters connected in seriesrdquo SmartMaterials and Structures vol 22 no 9 Article ID 094026 2013

[16] S-J Jang E Rustighi M J Brennan Y P Lee and H-JJung ldquoDesign of a 2DOF vibrational energy harvesting devicerdquoJournal of IntelligentMaterial Systems and Structures vol 22 no5 pp 443ndash448 2011

[17] L Tang and Y Yang ldquoA multiple-degree-of-freedom piezoelec-tric energy harvesting modelrdquo Journal of Intelligent MaterialSystems and Structures vol 23 no 14 pp 1631ndash1647 2012

[18] Z Yang and J Yang ldquoConnected vibrating piezoelectricbimorph beams as a wide-band piezoelectric power harvesterrdquoJournal of IntelligentMaterial Systems and Structures vol 20 no5 pp 569ndash574 2009

[19] W Zhou G R Penamalli and L Zuo ldquoAn efficient vibrationenergy harvester with a multi-mode dynamic magnifierrdquo SmartMaterials and Structures vol 21 no 1 Article ID 015014 2012

[20] D S Nguyen E Halvorsen G U Jensen and A VoglldquoFabrication and characterization of a widebandMEMS energyharvester utilizing nonlinear springsrdquo Journal of Micromechan-ics andMicroengineering vol 20 no 12 Article ID 125009 2010

[21] S D Nguyen and E Halvorsen ldquoNonlinear springs forbandwidth-tolerant vibration energy harvestingrdquo Journal ofMicroelectromechanical Systems vol 20 no 6 pp 1225ndash12272011

[22] S BoisseauGDespesse andBA Seddik ldquoNonlinear h-shapedsprings to improve efficiency of vibration energy harvestersrdquo

Journal of Applied Mechanics vol 80 no 6 Article ID 0610132013

[23] S Priya ldquoAdvances in energy harvesting using low profilepiezoelectric transducersrdquo Journal of Electroceramics vol 19 no1 pp 165ndash182 2007

[24] S Roundy P K Wright and J Rabaey ldquoA study of lowlevel vibrations as a power source for wireless sensor nodesrdquoComputer Communications vol 26 no 11 pp 1131ndash1144 2003

[25] S Roundy and P K Wright ldquoA piezoelectric vibration basedgenerator for wireless electronicsrdquo Smart Materials and Struc-tures vol 13 no 5 pp 1131ndash1142 2004

[26] H A Sodano G Park and D J Inman ldquoEstimation of electriccharge output for piezoelectric energy harvestingrdquo Strain vol40 no 2 pp 49ndash58 2004

[27] N E Dutoit B L Wardle and S-G Kim ldquoDesign considera-tions for mems-scale piezoelectric mechanical vibration energyharvestersrdquo Integrated Ferroelectrics vol 71 no 1 pp 121ndash1602005

[28] S-N Chen G-J Wang andM-C Chien ldquoAnalytical modelingof piezoelectric vibration-induced micro power generatorrdquoMechatronics vol 16 no 7 pp 379ndash387 2006

[29] A Erturk and D J Inman ldquoIssues in mathematical modelingof piezoelectric energy harvestersrdquo Smart Materials and Struc-tures vol 17 no 6 Article ID 065016 2008

[30] F Lu H P Lee and S P Lim ldquoModeling and analysis of micropiezoelectric power generators for micro-electromechanical-systems applicationsrdquo Smart Materials and Structures vol 13no 1 pp 57ndash63 2004

[31] S Jiang X Li S Guo Y Hu J Yang andQ Jiang ldquoPerformanceof a piezoelectric bimorph for scavenging vibration energyrdquoSmartMaterials and Structures vol 14 no 4 pp 769ndash774 2005

[32] A Erturk and D J Inman ldquoA distributed parameter elec-tromechanical model for cantilevered piezoelectric energy har-vestersrdquo Journal of Vibration and Acoustics vol 130 no 4Article ID 041002 2008

[33] A Erturk and D J Inman ldquoAn experimentally validatedbimorph cantilever model for piezoelectric energy harvestingfrom base excitationsrdquo Smart Materials and Structures vol 18no 2 Article ID 025009 2009

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Submit your manuscripts athttpwwwhindawicom

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Propagation




Navigation and Observation



DistributedSensor Networks



Mt Rl (t)

+

minus


(a)

Mt Rl (t)

+

minus


(b)

Figure 1 Cantilever piezoelectric energy configurations (a) unimorph (b) bimorph (parallel connection)

an operating frequency equal to their resonance frequencyand if the excitation is slightly different there is a significantdrop in the level of power generated This problem has ledto the search for broadband energy harvesters Twiefel andWestermann [10] present a survey of broadband techniquesfor vibration-based energy harvesting These techniques areclassified as follows

(i) Linear generators arrays combined modes

(ii) Nonlinear generators bimultiple stable vibroimpactresonant tuning

(iii) Advanced electronic networks switching networkssynthetic impedance

According to the researchers there is no universal solutiondepending on the application one technique may be betterthan the others For example linear generators are moreefficient for stochastic excitations whereas nonlinear tech-niques are more suitable for harmonic excitations or very lowexcitation frequencies

In the case of linear generators recent research hasbeen focused on developing systems that exhibit multipleresonances that are closely spaced The simplest solution isto use multiple cantilevers The cantilevers are built withdifferent geometries (large tip mass or width) so each beamhas a slightly different resonant frequency thus increasing thebandwidth of the full system A first design of a cantileverarray was presented by Shahruz [11 12] He concluded thatif the dimensions of the beams and masses are chosenappropriately then the system performs as a broadbandharvester However the maximal frequency band is limitedand is independent of the beams dimensions This approachhas been studied extensively since then [13ndash15]

Another method is to use structures with two or moredegrees of freedom resulting in two or more resonantfrequencies that are closely spaced [16ndash19] Z Yang andJ Yang [18] analyzed the power generated by the flexuralvibrations of two piezoelectric beams connected by springsthey showed that this structure can be used as a broadbandenergy harvester Nevertheless the model presented by theauthors is restricted to two beams and the extension tomore than two beams is not trivial Springs have also beenused in MEMS-based energy harvesters where it has been

demonstrated that the use of nonlinear springs increases thegenerated power [20ndash22]

The research community related to piezoelectric energyharvesting is very wide and includes researchers frommechanical electrical materials and civil engineering Thishas led to the development of many different models for thesame problem a piezoelectric cantilevered beam Most ofthese models use simplifications such as oversimplifying ofthe piezoelectric coupling as viscous damping [23] using ofthe static deflection of the beam in a dynamic model [24 25]no inclusion of the piezoelectric effect in the mechanicalmodel [26] or approximate solutions as the Rayleigh-Ritzapproximation [27 28] Erturk and Inman [29] providea comprehensive discussion of the different simplificationsfound in literature to model piezoelectric beams highlight-ing the common indiscretions and providing correctionsand necessary clarifications for researchers from differentengineering fields The best solution can be obtained bysolving the continuous electromechanical model analyticallyDifferent approaches for solving the continuousmodel can befound in literature [30ndash33]

In this paper we present a general approach to character-ize multiple piezoelectric beams connected with springs Theanalytical solution of the cantilevered piezoelectric energyharvester with Euler-Bernoulli beam assumptions proposedby Erturk and Inman [33] is extended to the case of an arrayof piezoelectric beams connected by springs The proposedconfiguration is attractive since it combines the properties ofbeam arrays and of systems withmultiple degrees of freedomFurthermore the results demonstrate that connecting anarray of beams with springs allows increasing the frequencyband of operation and increasing the amount of powergenerated when compared with an array of disconnectedbeams

The remainder of this work is organized as followsSection 2 presents the structure under investigation andthe methodology to compute its modes shapes and naturalfrequencies Section 3 provides the derivation of the elec-tromechanical equations of a bimorph cantilever beam InSection 4 the equations of the output voltage and power ofan array of bimorph cantilever beams connected by springsare derived Section 5 presents analytical and experimentalresults for different combinations of tip masses and springsconstants comparing the results with those of an array ofdisconnected beams (with no springs) Finally conclusionsand forthcoming work are presented in Section 6


Mt1

Mt2

MtN

k1

k2

kN

(a) Full system

kiminus1

kiwi(x t)

Mtiminus1

Mti

Mti+1

X






119894(119909 119905) =

120601119894(119909)120578(119905) The function 120601



119894(119909) is

120601119894 (119909) = 120582

119894

1sin (120573119909) + 120582119894

2cos (120573119909) + 120582119894

3sinh (120573119909)

+ 120582119894

4cosh (120573119909)

(1)


120601119894 (0) = 0

120597120601119894 (0)

120597119909= 0

(2)

which implies that

120582119894

4= minus120582119894

2

120582119894

3= minus120582119894

1

(3)


120601119894 (119909) = 120582

119894

1[sin (120573119909) minus sinh (120573119909)]

+ 120582119894

2[cos (120573119909) minus cosh (120573119909)]

(4)


1198841198681205972119908119894

1205971199092= 0

120597

120597119909[119884119868

1205972119908119894

1205971199092] = minus119872

119905119894

1205972119908119894

1205971199052minus 119896119894(119908119894minus 119908119894+1)


(5)



119894is the


119894(119909)120578(119905)

and 1205972120578(119905)1205971199052 = minus1205962119899120578(119905) 120596


yields

1205972120601119894 (119871)

1205971199092= 0

1198841198681205973120601119894 (119871)

1205971199093= 1205962

119899119872119905119894120601119894 (119871) minus 119896119894 [120601119894 (119871) minus 120601119894+1 (119871)]


(6)



[

0 0 1205721

1205722

0 0


[[[[[[[[[[[[

[

120582119894minus1

1

120582119894minus1

2

120582119894

1

120582119894

2

120582119894+1

1

120582119894+1

2

]]]]]]]]]]]]

]

=

[[[[[[[[[[[

[

0

0

0

0

0

0

]]]]]]]]]]]

]

(7)

with

1205721= sin (120573119871) + sinh (120573119871)

1205722= cos (120573119871) + cosh (120573119871)

1205723(119896119894) = 119896119894(sinh (120573119871) minus sin (120573119871))

1205724(119896119894) = 119896119894(cosh (120573119871) minus cos (120573119871))

1205725(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (sin (120573119871) minus sinh (120573119871))

+ 1198841198681205733(cos (120573119871) + cosh (120573119871))

1205726(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (cos (120573119871) minus cosh (120573119871))

+ 1198841198681205733(sinh (120573119871) minus sin (120573119871))

(8)


[

1205721

1205722

0 0

1205725(1198721199051 1198961 0) 120572

6(1198721199051 1198961 0) 120572

3(1198961) 1205724(1198961)]

[[[[[[

[

1205821

1

1205821

2

1205822

1

1205822

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(9)


[

0 0 1205721

1205722


[[[[[[

[

120582119873minus1

1

120582119873minus1

2

120582119873

1

120582119873

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(10)


C120582 = 0 (11)


2 1205822

1 1205822

2

120582119873

1 120582119873



119903 The




119903is obtained

Given 120573119903and 120582


120601119903119894 (119909) = 120582

119894

1199031[sin (120573

119903119909) minus sinh (120573

119903119909)]

+ 120582119894

1199032[cos (120573

119903119909) minus cosh (120573

119903119909)]

(12)


120596119899119903= 1205732

119903radic119884119868

119898 (13)



120573rL = 1536

(a) Mode 1

120573rL = 1573

(b) Mode 2

120573rL = 1640

(c) Mode 3

120573rL = 4192




=

03119898119871 11989612= 03119864119868119871

3)




1205972119872(119909 119905)

1205971199092+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(14)

b

x(t)

Mt

hp

hp

hs

PZT

PZTSubstructureY0e

i120596t








function



119872(119909 119905) = minus119887 [int

minusℎ1199042


119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)




119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)


119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)



1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901



119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)


119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)



119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)


120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)


1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)


119863 = 11988931119879119901+ 120576119879

33119864 (25)



119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879



119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ



119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)


119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)



119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Mt1

Mt2

MtN

k1

k2

kN

(a) Full system

kiminus1

kiwi(x t)

Mtiminus1

Mti

Mti+1

X






119894(119909 119905) =

120601119894(119909)120578(119905) The function 120601



119894(119909) is

120601119894 (119909) = 120582

119894

1sin (120573119909) + 120582119894

2cos (120573119909) + 120582119894

3sinh (120573119909)

+ 120582119894

4cosh (120573119909)

(1)


120601119894 (0) = 0

120597120601119894 (0)

120597119909= 0

(2)

which implies that

120582119894

4= minus120582119894

2

120582119894

3= minus120582119894

1

(3)


120601119894 (119909) = 120582

119894

1[sin (120573119909) minus sinh (120573119909)]

+ 120582119894

2[cos (120573119909) minus cosh (120573119909)]

(4)


1198841198681205972119908119894

1205971199092= 0

120597

120597119909[119884119868

1205972119908119894

1205971199092] = minus119872

119905119894

1205972119908119894

1205971199052minus 119896119894(119908119894minus 119908119894+1)


(5)



119894is the


119894(119909)120578(119905)

and 1205972120578(119905)1205971199052 = minus1205962119899120578(119905) 120596


yields

1205972120601119894 (119871)

1205971199092= 0

1198841198681205973120601119894 (119871)

1205971199093= 1205962

119899119872119905119894120601119894 (119871) minus 119896119894 [120601119894 (119871) minus 120601119894+1 (119871)]


(6)



[

0 0 1205721

1205722

0 0


[[[[[[[[[[[[

[

120582119894minus1

1

120582119894minus1

2

120582119894

1

120582119894

2

120582119894+1

1

120582119894+1

2

]]]]]]]]]]]]

]

=

[[[[[[[[[[[

[

0

0

0

0

0

0

]]]]]]]]]]]

]

(7)

with

1205721= sin (120573119871) + sinh (120573119871)

1205722= cos (120573119871) + cosh (120573119871)

1205723(119896119894) = 119896119894(sinh (120573119871) minus sin (120573119871))

1205724(119896119894) = 119896119894(cosh (120573119871) minus cos (120573119871))

1205725(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (sin (120573119871) minus sinh (120573119871))

+ 1198841198681205733(cos (120573119871) + cosh (120573119871))

1205726(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (cos (120573119871) minus cosh (120573119871))

+ 1198841198681205733(sinh (120573119871) minus sin (120573119871))

(8)


[

1205721

1205722

0 0

1205725(1198721199051 1198961 0) 120572

6(1198721199051 1198961 0) 120572

3(1198961) 1205724(1198961)]

[[[[[[

[

1205821

1

1205821

2

1205822

1

1205822

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(9)


[

0 0 1205721

1205722


[[[[[[

[

120582119873minus1

1

120582119873minus1

2

120582119873

1

120582119873

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(10)


C120582 = 0 (11)


2 1205822

1 1205822

2

120582119873

1 120582119873



119903 The




119903is obtained

Given 120573119903and 120582


120601119903119894 (119909) = 120582

119894

1199031[sin (120573

119903119909) minus sinh (120573

119903119909)]

+ 120582119894

1199032[cos (120573

119903119909) minus cosh (120573

119903119909)]

(12)


120596119899119903= 1205732

119903radic119884119868

119898 (13)



120573rL = 1536

(a) Mode 1

120573rL = 1573

(b) Mode 2

120573rL = 1640

(c) Mode 3

120573rL = 4192




=

03119898119871 11989612= 03119864119868119871

3)




1205972119872(119909 119905)

1205971199092+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(14)

b

x(t)

Mt

hp

hp

hs

PZT

PZTSubstructureY0e

i120596t








function



119872(119909 119905) = minus119887 [int

minusℎ1199042


119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)




119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)


119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)



1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901



119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)


119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)



119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)


120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)


1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)


119863 = 11988931119879119901+ 120576119879

33119864 (25)



119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879



119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ



119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)


119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)



119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











[

0 0 1205721

1205722

0 0


[[[[[[[[[[[[

[

120582119894minus1

1

120582119894minus1

2

120582119894

1

120582119894

2

120582119894+1

1

120582119894+1

2

]]]]]]]]]]]]

]

=

[[[[[[[[[[[

[

0

0

0

0

0

0

]]]]]]]]]]]

]

(7)

with

1205721= sin (120573119871) + sinh (120573119871)

1205722= cos (120573119871) + cosh (120573119871)

1205723(119896119894) = 119896119894(sinh (120573119871) minus sin (120573119871))

1205724(119896119894) = 119896119894(cosh (120573119871) minus cos (120573119871))

1205725(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (sin (120573119871) minus sinh (120573119871))

+ 1198841198681205733(cos (120573119871) + cosh (120573119871))

1205726(119872119905119894 119896119894 119896119894minus1)

= (119896119894+ 119896119894minus1minus1198721199051198941205962

119899) (cos (120573119871) minus cosh (120573119871))

+ 1198841198681205733(sinh (120573119871) minus sin (120573119871))

(8)


[

1205721

1205722

0 0

1205725(1198721199051 1198961 0) 120572

6(1198721199051 1198961 0) 120572

3(1198961) 1205724(1198961)]

[[[[[[

[

1205821

1

1205821

2

1205822

1

1205822

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(9)


[

0 0 1205721

1205722


[[[[[[

[

120582119873minus1

1

120582119873minus1

2

120582119873

1

120582119873

2

]]]]]]

]

=

[[[[[

[

0

0

0

0

]]]]]

]

(10)


C120582 = 0 (11)


2 1205822

1 1205822

2

120582119873

1 120582119873



119903 The




119903is obtained

Given 120573119903and 120582


120601119903119894 (119909) = 120582

119894

1199031[sin (120573

119903119909) minus sinh (120573

119903119909)]

+ 120582119894

1199032[cos (120573

119903119909) minus cosh (120573

119903119909)]

(12)


120596119899119903= 1205732

119903radic119884119868

119898 (13)



120573rL = 1536

(a) Mode 1

120573rL = 1573

(b) Mode 2

120573rL = 1640

(c) Mode 3

120573rL = 4192




=

03119898119871 11989612= 03119864119868119871

3)




1205972119872(119909 119905)

1205971199092+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(14)

b

x(t)

Mt

hp

hp

hs

PZT

PZTSubstructureY0e

i120596t








function



119872(119909 119905) = minus119887 [int

minusℎ1199042


119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)




119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)


119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)



1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901



119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)


119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)



119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)


120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)


1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)


119863 = 11988931119879119901+ 120576119879

33119864 (25)



119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879



119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ



119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)


119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)



119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120573rL = 1536

(a) Mode 1

120573rL = 1573

(b) Mode 2

120573rL = 1640

(c) Mode 3

120573rL = 4192




=

03119898119871 11989612= 03119864119868119871

3)




1205972119872(119909 119905)

1205971199092+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(14)

b

x(t)

Mt

hp

hp

hs

PZT

PZTSubstructureY0e

i120596t








function



119872(119909 119905) = minus119887 [int

minusℎ1199042


119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)




119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)


119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)



1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901



119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)


119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)



119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)


120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)


1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)


119863 = 11988931119879119901+ 120576119879

33119864 (25)



119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879



119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ



119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)


119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)



119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119872(119909 119905) = minus119887 [int

minusℎ1199042


119879119901119910119889119910 + int

ℎ1199042

minusℎ1199042

119879119904119910119889119910

+ int

ℎ1199042+ℎ119901

ℎ1199042

119879119901119910119889119910]

(15)




119879119904= 119884119904119878119904 (16)

119879119901= 119884119901119878119901minus 11988931119884119901119864 (17)


119878 (119909 119910 119905) = minus1199101205972119908rel (119909 119905)

1205971199092 (18)



1198641 (119905) = minus

V (119905)ℎ119901

in the top layer

1198642 (119905) =

V (119905)ℎ119901



119872(119909 119905) =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)]

sdot1205972119908rel (119909 119905)

1205971199092

+11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] V (119905)

(20)


119872(119909 119905) = 1198841198681205972119908rel (119909 119905)

1205971199092

+ 120599V (119905) [Φ (119909) minus Φ (119909 minus 119871)] (21)



119884119868 =2119887

3[119884119904ℎ3

119904

12+ 119884119901((ℎ119901+ℎ119904

2)

3

minusℎ3

119904

8)] (22)


120599 =11988931119884119901119887

ℎ119901

[ℎ2

119904

4minus (ℎ119901+ℎ119904

2)

2

] (23)


1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599V (119905) [120597120575 (119909)

120597119909minus120597120575 (119909 minus 119871)

120597119909]

= minus [119898 +119872119905120575 (119909 minus 119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(24)


119863 = 11988931119879119901+ 120576119879

33119864 (25)



119863(119909 119905) = 11988931 (119884119901119878119901minus 11988931119864) + 120576

119879

33119864

= 11988931119884119901119878119901+ 119864 (120576

119879

33minus 1198892

31119884119901)

= 11988931119884119901119878119901minus 120576119878

33

V (119905)ℎ119901

(26)

with 12057611987833= 1198892

31119884119901minus 120576119879



119878119901= minusℎ119901119888

1205972119908rel (119909 119905)

1205971199092 (27)

where ℎ119901119888= ℎ1199042 + ℎ



119863 (119909 119905) = minus11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092minus 120576119878

33

V (119905)ℎ119901

(28)


119902 (119905) = minusint

119871

0

(11988931119884119901ℎ119901119888

1205972119908rel (119909 119905)

1205971199092+ 120576119878

33

V (119905)ℎ119901

)119887119889119909 (29)



119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894 (119905) =120597119902 (119905)

120597119905

= minus120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(30)


119894 (119905) = 1198941 (119905) + 1198942 (119905)

= minus2120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 2int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(31)



V (119905) = minus2119877119897 [120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

+ int

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909]

(32)


V (119905)2119877119897

+120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

= minusint

119871

0

(11988931119884119901ℎ1199011198881198871205973119908rel (119909 119905)

1205971199092120597119905) 119889119909

(33)



1198841198681205974119908rel (119909 119905)

1205971199094+ 1198881199041198681205975119908rel (119909 119905)

1205971199094120597119905+ 119888119886

120597119908rel (119909 119905)

120597119905

+ 1198981205972119908rel (119909 119905)

1205971199052

+ 120599

119873

sum

119894=1

V119894 (119905) [

120597120575 (119909 minus (119894 minus 1) 119871)

120597119909minus120597120575 (119909 minus 119894119871)

120597119909]

= minus[119898 +

119873

sum

119894=1

119872119905119894120575 (119909 minus 119894119871)]

1205972119908119887 (119905)

1205971199052minus 119888119886

120597119908119887 (119905)

120597119905

(34)

(t)Rl




1(119905) = V





119908rel (119909 119905) = sum119903

120601119903 (119909) 120578119903 (119905) (35)

where 120601119903(119909) and 120578



120601119903 (119909)

=

1206011199031 (119909) 0 le 119909 le 119871

1206011199032 (119909 minus 119871) 119871 le 119909 le 2119871


(36)

where 120601119903119894



[1205962

119903120578119903 (119905) + 2120585119903120596119903 120578119903 (119905) + 120578

119903 (119905)] 120572119903 +

119873

sum

119894=1

120594119903119894V (119905)

= [minus1198981205972119908119887 (119909 119905)

1205971199052minus 119888119886

120597119908119887 (119909 119905)

120597119905] 120574119903

minus1205972119908119887 (119909 119905)

1205971199052

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(37)


with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










with

120594119903119894= 120599(minus120601

119903119894 (119871) +120597120601119903119894 (119871)

120597119909)

120585119903=119888119904119868120596119903

2119884119868+

119888119886

2119898120596119903

120572119903= int

119873119871

0

1198981206012

119903(119909) 119889119909

120574119903= int

119873119871

0

120601119903 (119909) 119889119909

(38)


0119890119895120596119905 and


119903119890119895120596119905 and V(119905) = 119881119890119895120596119905 (37)

becomes

[1205962

119899119903119873119903+ 2119895120585119903120596119899119903120596119873119903minus 1205962119873119903] 120572119903+ 119881

119873

sum

119894=1

120594119903119894

= (1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840

119873

sum

119894=1

119872119905119894120601119903 (119894119871)

(39)



119873119903

=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871) minus 119881sum

119873

119894=1120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(40)


119873119903= 119886119903minus 119881

119873

sum

119894=1

119887119903119894 (41)

with

119886119903=

(1198981205962minus 119895120596119888119886) 1198840120574119903+ 12059621198840sum119873

119894=1119872119905119894120601119903 (119894119871)

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

119887119903119894=

120594119903119894

(1205962119903+ 2119895120585119903120596119903120596 minus 1205962) 120572

119903

(42)


119908rel119894 (119909 119905) = sum119903

120601119903119894 (119909) 120578119903 (119905) (43)


119894119894 (119905) = minus2

120576119878

33119887119871

ℎ119901

120597V (119905)120597119905

minus 211988931119884119901ℎ119901119888119887sum

119903

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(44)


119897119894(119905) = 119877



V (119905)2119877119897

= minus120576119878

33119887119871119873

ℎ119901

120597V (119905)120597119905

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120578119903 (119905)

(45)


119881

2119877119897

= minus119895120596120576119878

33119887119871119873

ℎ119901

119881

minus 11988931119884119901ℎ119901119888119887sum

119903

sum

119894

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

119895120596119873119903

(46)


119881120581119888= sum

119903

sum

119894

120593119903119894119895120596119873119903 (47)

with

120593119903119894= minus

11988931119884119901ℎ119901119888ℎ119901

120576119878

33119871119873

120597120601119903119894 (119909)

120597119909

10038161003816100381610038161003816100381610038161003816

119871

0

120581119888= 119895120596 +

ℎ119901

2119877119897120576119878

33119887119871119873

(48)


obtained

119881120581119888= sum

119903

sum

119894

120593119903119894119895120596(119886

119903minus 119881

119873

sum

119896=1

119887119903119896) (49)


119881 =sum119903sum119894120593119903119894119895120596119886119903

120581119888+ sum119903sum119894120593119903119894119895120596sum119873

119896=1119887119903119896

(50)


0 The instantaneous




5 Results





One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(a) Connected

One beamTwo beams

Four beamsSix beams

50 100 150 200

Frequency (Hz)

10minus2

100

10minus4

10minus6

Pow

er F

RF (W

s4m

2)

(b) Disconnected


= 03119898119871 and 119896119894=15

= 031198841198681198713

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(a) Connected

Two beamsFour beams

Six beams

50 100 150 200

Frequency (Hz)

10minus4

10minus3

10minus2

10minus1

100

Pow

er F

RF (W

s4m

2)

(b) Disconnected


1199052= 035119898119871

119896 = 031198841198681198713 three beams119872

1199051= 025119898119871119872

1199052= 030119898119871119872

1199053= 035119898119871 119896

12= 03119884119868119871

3 six beams1198721199051= 015119898119871119872

1199052= 020119898119871

1198721199053= 030119898119871119872

1199054= 035119898119871119872

1199055= 040119898119871119872

1199056= 045119898119871 119896

119894=15= 03119884119868119871

3



119904(mm) 019















50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

01mL 02mL01mL 03mL

01mL 04mL01mL 05mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


k = 04YIL3



50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100


03mL 03mL 04mL


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

k =


02YIL3 06YIL3







1199051= 65119898119871119872

1199052= 35119898119871


1= 1198962=

2921198841198681198713


50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected02YIL3

(a) 119896119894=19 = 021198841198681198713

50 100 150 200

Frequency (Hz)

10minus4

Pow

er F

RF (W

s4m

2)

10minus3

10minus2

10minus1

100

Connected k =

Disconnected03YIL3

(b) 119896119894=19 = 031198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected04YIL3

(c) 119896119894=19 = 041198841198681198713

50 100 150 200

Frequency (Hz)

Pow

er F

RF (W

s4m

2)

10minus4

10minus3

10minus2

10minus1

100

Connected k =

Disconnected05YIL3

(d) 119896119894=19 = 051198841198681198713


1199052= 015119898119871119872

1199053= 02119898119871119872

1199054= 025119898119871119872

1199055= 03119898119871119872

1199056= 035119898119871

1198721199057= 04119898119871119872

1199058= 045119898119871119872

1199059= 05119898119871119872

11990510= 055119898119871







Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)

Beam 1Beam 2

Beam 3

Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(a) Experimental

20 25 30 35 40 45 50 55 60 65 70

Frequency (Hz)


Pow

er F

RF (W

s4m

2)

10minus7

10minus6

10minus9

10minus8

10minus5

10minus4

(b) Analytical


6 Conclusions






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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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Research Article A Broadband Vibration-Based Energy ...downloads.hindawi.com/journals/sv/2016/9614842.pdf · a broadband vibration-based energy harvester. e equations for the voltage