Osman Sayginer · For anlytical calculations, cantilever is broken20 nodes. And then all; global...
1
Physics of Microcantilever Introduction Beam Theory and Mathematical Modelling Acknowledgments • This project is being supported by The Scientific And Technological Research Council of Turkey (TUBITAK) within 2209/A - 2014/2 Research Founding Programme for undergraduate students. Results Conclusions References • Suter, M., et al. "Superparamagnetic photocurable nanocomposite for the fabrication of microcantilevers." Journal of Micromechanics and Microengineering 21.2 (2011): 025023. • Boisen, Anja, et al. "Cantilever-like micromechanical sensors." Reports on Progress in Physics 74.3 (2011): 036101. • Jiang, Lei, Yuejuan Li, and Cheng Marvin. "Analysis of piezoelectric energy harvesting device with adjustable resonance frequency." Journal of Electronics (China) 29.3-4 (2012): 310-318. • Hatch, Michael R. Vibration simulation using MATLAB and ANSYS. CRC Press, 2012. • Young W C, Budynas R G and Roark R J 2002 Roark’s Formulas for Stress and Strain (Boston, A: McGraw-Hill). • Dutoit, Noël E., Brian L. Wardle, and Sang-Gook Kim. "Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters." Integrated Ferroelectrics 71.1 (2005): 121-160. Euler-Bernoulli or thin beam theory is considered. General equation of motion or the beam structure is given below. 2 (, ) 2 Г + 4 (, ) 4 =0 Where , is the displacement in the x-direction, is the mass density, Г = is the cross-sectional area, is Young’s modulus and is the geometric moment of inertia. The solution U(, )= ()exp(−2 ), here is the frequency of motion and where to this differential equation is a harmonic that can be separated into a position dependent and a time-dependent term, denotes the modal number. 4 (, ) 4 = 4 , , 4 = 2 Г The solutions (eigenfunctions) to the simplified beam equation can be written in the form; = ( − )+ ( − ) The modal constants are determined from the boundary conditions. For a singly clamped beam (cantilever), the frequency equation becomes 1+ =0 Solutions for = are shown below: Microfabricated micro cantilevers are widely used as a nanomechanical tool for diagnostics, molecular detections and many sensor devices. Its easiness, sensivity and cost of manufacture makes it a key element for many applications. Piezoelectric material covered unimorph microcantilevers would provide energy generation from ambient sources as a result of its freely vibrating body. To generate maximum amount of energy, the microcantilever structure’s resonance frequency (natural frequency) should have be specifically adjusted for energy sources. However resonance frequency is limited by cantilever’s dimensions and materials. Consciously designed cantilevers will be able to harvest maximum amount of energy from ambient sources. Hence resonance frequency of double layered unimorph cantilever should have be investigated. In this study we have represented and demonstrated the finite element method (FEM) to understand the effect of thickness and length parameters on resonance frequencies of silicon based on microcantilever beam. A thin layer of Lead Zirconate Titanate (PZT-5H) was integrated on rectangular silicon proof mask cantilever as piezoelectric material. COMSOL Multiphysics which is commercial FEM Software, is used for FEM analysis. Mode shapes and mode frequencies are obtained. Then results are compared with Euler-Bernoulli based on analytical calculation results which are done by MATLAB. SEM image of a microcantilever. SEM image of MEMS piezoelectric vibration energy harvester. «Harvesting of vibration energy from mechanical operations or human motions is getting more popular due to its broad applications and power density (~ 200 µW/ cm3)» = 2 2 2 12 4 Where is Elastic Modulus, is width and is height of cantilever, is thickness of the cantilever, is length of the cantilever and is density the cantilever. Where m is mass of the cantilever. = 2 4 + 2 4 +2 (2 2 + 2 2 +3 ) 12( + ) = 3 3 For a unimorph cantilever the spring mass is extracted according to bending modulus ( ), thickness ( ) and Elastic Modulus ( ) of piezoelectric layer, thickness ( ) and Elastic Modulus ( ) of elastic layer: Natural frequency also can be rewritten in the following form: = 2 2 3 Layer / Material Length (µ) Width (µ) Thickness (µ) Elastic Modulus () Density ( � ) Elastic / Silicon 400 64 8 170 2329 Piezoelectric / Lead Zirconate Titanate (PZT-5H) 400 64 2 63 7500 Cantilever properties are given below. For anlytical calculations, cantilever is broken 20 nodes. And then all global stiffness and mass matrices are solved using MATLAB reffering. Maximum displacement of cantilever is also calculated which is not presented here. Relations between length,thickhness and mode frequencies are shown below. 18 23 28 33 38 43 48 53 58 63 400 425 450 475 500 525 550 575 600 625 650 675 700 1st Mode Frequency (kHz) Length (µm) COMSOL MATLAB 60 61 62 63 64 65 66 67 2 3 4 5 6 1st Mode Natural Frequency (kHz) Thickness of Piezoelectric Layer (μm) COMSOL MATLAB 60 65 70 75 80 85 90 95 8 9 10 11 12 1st Mode Natural Frequency (kHz) Thickness of Elastic Layer (µm) COMSOL MATLAB From left to right, the first graph shows Natural Frequency decreases while increasing length of cantilever. Those results are same as expected from natural frequency formula on the left column. Second and third graphs show natural frequency increases while decreasing thichness of cantilever. Key point of between second and third graph is, piezoelectric layer is thinner and also softer than elastic layer. So thickness of the elastic layer effects the caltilever’s frequency on a large scale. Mod Shapes and Frequencies 0 500 1000 1500 2000 2500 0 1 2 3 4 Mode Frequencies (kHz) Modes COMSOL MATLAB Second and upper mode frequencies decreases extremly comparing first mode because of the mode number. This study presents simulation studies of unimorph piezoelectric beam structure for energy harvesting applications. The derivations of the mathematical equations are based on Euler-Bernoulli beam theory for analytical calculations. For unimorph structures natural frequency formula is written regarding to bending modulus and spring constant variables. In this study we also have presented resonance frequencies and mode shapes for first 6 modes. As shown graphs before we discussed cantilever’s dimentions and their effect on resonance frequency. Thickness of cantilever is directly proportional to natural frequency. However cantilever’s length is inversly proportional to the natural frequency. Natural frequency formula is only accurate for first mode frequency because of its linearity. Therefore FEM analysis are very important and necessary for accurate estimation. Elastic Layer Piezoelectric Layer Schematic of a simple cantilever geometry. Natural (resonance) frequency ( ) and spring constant ( ) of cantilever are expressed below. 1 =61 5 =1070 6 =2090 2 =383 3 =421 4 =727 1 2 3 4 5 6 1.875 4.694 7.855 10.996 14.137 17.279 Osman Sayginer Namik Kemal University, Dept . of Mechanical Engineering www.sayginer.com Recently, Dagdeviren et al.* have announced new energy harvesting approach which aims to generating energy for pacemakers. They have developed a PZT cantilever based on device that uses the motion of heartbeats (as well as the motion of the lungs and diaphragm) to power an implanted pacemaker. *Conformal piezoelectric energy harvesting and storage from motions of the heart, lung, and diaphragm." National Academy of Sciences (2014)
Transcript of Osman Sayginer · For anlytical calculations, cantilever is broken20 nodes. And then all; global...
Physics of Microcantilever
Introduction
Beam Theory and Mathematical Modelling
Acknowledgments• This project is being supported by The Scientific And Technological Research Council of Turkey (TUBITAK) within 2209/A - 2014/2 Research Founding Programme for
undergraduate students.
Results
Conclusions
References• Suter, M., et al. "Superparamagnetic photocurable nanocomposite for the fabrication of
microcantilevers." Journal of Micromechanics and Microengineering 21.2 (2011): 025023.• Boisen, Anja, et al. "Cantilever-like micromechanical sensors." Reports on Progress in Physics
74.3 (2011): 036101.• Jiang, Lei, Yuejuan Li, and Cheng Marvin. "Analysis of piezoelectric energy harvesting device
with adjustable resonance frequency." Journal of Electronics (China) 29.3-4 (2012): 310-318.
• Hatch, Michael R. Vibration simulation using MATLAB and ANSYS. CRC Press, 2012.• Young W C, Budynas R G and Roark R J 2002 Roark’s Formulas for Stress and Strain (Boston,
A: McGraw-Hill).• Dutoit, Noël E., Brian L. Wardle, and Sang-Gook Kim. "Design considerations for MEMS-scale
piezoelectric mechanical vibration energy harvesters." Integrated Ferroelectrics 71.1 (2005): 121-160.
Euler-Bernoulli or thin beam theory is considered. General equation of motion or the beam structure is given below.
𝜕𝜕2𝑈𝑈(𝑧𝑧, 𝑡𝑡)𝜕𝜕𝑡𝑡2
Г𝜌𝜌 +𝜕𝜕4𝑈𝑈(𝑧𝑧, 𝑡𝑡)𝜕𝜕𝑧𝑧4
𝐸𝐸𝐸𝐸 = 0
Where 𝑈𝑈 𝑧𝑧, 𝑡𝑡 is the displacement in the x-direction, 𝜌𝜌 is the mass density, Г =𝑤𝑤𝑤 is the cross-sectional area, 𝐸𝐸 is Young’s modulus and 𝐸𝐸 is the geometricmoment of inertia.The solution U(𝑧𝑧, 𝑡𝑡) = 𝑈𝑈𝑛𝑛(𝑧𝑧)exp(−𝑖𝑖2𝜋𝜋𝑓𝑓𝑛𝑛𝑡𝑡), here 𝑓𝑓𝑛𝑛 is the frequency of motionand where 𝑛𝑛to this differential equation is a harmonic that can be separated intoa position dependent and a time-dependent term, 𝑛𝑛 denotes the modal number.
𝐸𝐸𝐸𝐸𝜕𝜕4𝑈𝑈(𝑧𝑧, 𝑡𝑡)𝜕𝜕𝑧𝑧4
= 𝑘𝑘4𝑈𝑈 𝑧𝑧, 𝑡𝑡 , 𝑘𝑘4 =𝜔𝜔 2Г𝜌𝜌𝐸𝐸𝐸𝐸
The solutions (eigenfunctions) to the simplified beam equation can be written in the form;
𝑈𝑈𝑛𝑛 𝑧𝑧 = 𝐴𝐴𝑛𝑛(𝑐𝑐𝑐𝑐𝑐𝑐 𝜅𝜅𝑛𝑛𝑙𝑙 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑤 𝜅𝜅𝑛𝑛𝑙𝑙) + 𝐵𝐵𝑛𝑛(𝑐𝑐𝑖𝑖𝑛𝑛 𝜅𝜅𝑛𝑛𝑙𝑙 − 𝑐𝑐𝑖𝑖𝑛𝑛𝑤 𝜅𝜅𝑛𝑛𝑙𝑙)
The modal constants are determined from the boundary conditions. For a singlyclamped beam (cantilever), the frequency equation becomes
1 + 𝑐𝑐𝑐𝑐𝑐𝑐 𝜅𝜅𝑛𝑛𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐𝑤 𝜅𝜅𝑛𝑛𝑙𝑙 = 0
Solutions for 𝜆𝜆𝑛𝑛 = 𝜅𝜅𝑛𝑛𝑙𝑙 are shown below:
Microfabricated micro cantilevers are widely used as ananomechanical tool for diagnostics, molecular detectionsand many sensor devices. Its easiness, sensivity and cost ofmanufacture makes it a key element for many applications.Piezoelectric material covered unimorph microcantileverswould provide energy generation from ambient sources asa result of its freely vibrating body. To generate maximumamount of energy, the microcantilever structure’sresonance frequency (natural frequency) should have bespecifically adjusted for energy sources. Howeverresonance frequency is limited by cantilever’s dimensionsand materials.Consciously designed cantilevers will be able to harvestmaximum amount of energy from ambient sources. Henceresonance frequency of double layered unimorphcantilever should have be investigated.In this study we have represented and demonstrated thefinite element method (FEM) to understand the effect ofthickness and length parameters on resonance frequenciesof silicon based on microcantilever beam. A thin layer ofLead Zirconate Titanate (PZT-5H) was integrated onrectangular silicon proof mask cantilever as piezoelectricmaterial. COMSOL Multiphysics which is commercial FEMSoftware, is used for FEM analysis. Mode shapes andmode frequencies are obtained. Then results arecompared with Euler-Bernoulli based on analyticalcalculation results which are done by MATLAB.
SEM image of a microcantilever.
SEM image of MEMS piezoelectric vibration energy harvester.
«Harvesting of vibration energy from mechanical operations or human motions is getting more
popular due to its broad applications and power density
(~ 200 µW/ cm3)»
𝑓𝑓𝑛𝑛 =𝜆𝜆𝑛𝑛
2
2𝜋𝜋𝐸𝐸𝑤2
12𝜌𝜌𝑙𝑙4
Where 𝐸𝐸 is ElasticModulus, 𝑤𝑤 is width and𝑤 is height of cantilever, 𝑡𝑡is thickness of thecantilever, 𝑙𝑙 is length ofthe cantilever and 𝜌𝜌 isdensity the cantilever.
Where m is mass of the cantilever.
𝐷𝐷𝑝𝑝 =𝐸𝐸𝑝𝑝2𝑡𝑡𝑝𝑝4 + 𝐸𝐸𝑠𝑠2𝑡𝑡𝑠𝑠4 + 2𝐸𝐸𝑝𝑝𝐸𝐸𝑠𝑠𝑡𝑡𝑝𝑝𝑡𝑡𝑠𝑠(2𝑡𝑡𝑝𝑝2 + 2𝑡𝑡𝑠𝑠2 + 3𝑡𝑡𝑝𝑝𝑡𝑡𝑠𝑠)
12(𝐸𝐸𝑝𝑝𝑡𝑡𝑝𝑝 + 𝐸𝐸𝑠𝑠𝑡𝑡𝑠𝑠)𝑘𝑘 =
3𝐷𝐷𝑝𝑝𝑤𝑤𝑙𝑙3
For a unimorph cantilever the spring mass is extracted according to bendingmodulus (𝐷𝐷𝑝𝑝), thickness (𝑡𝑡𝑝𝑝) and Elastic Modulus (𝐸𝐸𝑝𝑝) of piezoelectric layer,thickness (𝑡𝑡𝑒𝑒) and Elastic Modulus (𝐸𝐸𝑒𝑒) of elastic layer:
Natural frequency also can be rewritten in the following form:
𝑓𝑓𝑛𝑛 =𝜆𝜆𝑛𝑛
2
2𝜋𝜋𝑘𝑘
3𝑚𝑚
Layer / Material Length (µ𝑚𝑚) Width (µ𝑚𝑚) Thickness (µ𝑚𝑚) Elastic Modulus (𝐺𝐺𝐺𝐺𝐺𝐺) Density ( �𝒌𝒌𝒌𝒌𝒎𝒎𝟑𝟑)
Elastic / Silicon 400 64 8 170 2329Piezoelectric / Lead Zirconate Titanate (PZT-5H)
400 64 2 63 7500
Cantilever properties are given below. For anlytical calculations, cantilever is broken 20 nodes. And then all global stiffness and massmatrices are solved using MATLAB reffering. Maximum displacement of cantilever is also calculated which is not presented here.
Relations between length,thickhness and mode frequencies are shown below.
18
23
28
33
38
43
48
53
58
63
400 425 450 475 500 525 550 575 600 625 650 675 700
1st M
ode
Freq
uenc
y (k
Hz)
Length (µm)
COMSOL MATLAB
60
61
62
63
64
65
66
67
2 3 4 5 6
1st M
ode
Nat
ural
Fre
quen
cy (k
Hz)
Thickness of Piezoelectric Layer (µm)
COMSOL MATLAB
60
65
70
75
80
85
90
95
8 9 10 11 12
1st M
ode
Nat
ural
Fre
quen
cy (k
Hz)
Thickness of Elastic Layer (µm)
COMSOL MATLAB
From left to right, the first graph shows Natural Frequency decreases while increasing length of cantilever. Those results are same asexpected from natural frequency formula on the left column.Second and third graphs show natural frequency increases while decreasing thichness of cantilever. Key point of between second andthird graph is, piezoelectric layer is thinner and also softer than elastic layer. So thickness of the elastic layer effects the caltilever’sfrequency on a large scale.
Mod Shapes and Frequencies
0
500
1000
1500
2000
2500
0 1 2 3 4
Mod
e Fr
eque
ncie
s (kH
z)
Modes
COMSOL MATLAB
Second and upper mode frequencies decreasesextremly comparing first mode because of the 𝑛𝑛mode number.
This study presents simulation studies of unimorphpiezoelectric beam structure for energy harvestingapplications. The derivations of the mathematicalequations are based on Euler-Bernoulli beam theoryfor analytical calculations. For unimorph structuresnatural frequency formula is written regarding tobending modulus and spring constant variables.
In this study we also have presented resonancefrequencies and mode shapes for first 6 modes. Asshown graphs before we discussed cantilever’sdimentions and their effect on resonance frequency.Thickness of cantilever is directly proportional tonatural frequency. However cantilever’s length isinversly proportional to the natural frequency.
Natural frequency formula is only accurate for firstmode frequency because of its linearity. ThereforeFEM analysis are very important and necessary foraccurate estimation.
𝑙𝑙𝑤𝑤
𝑡𝑡𝑝𝑝𝑡𝑡𝑒𝑒
Elastic Layer
Piezoelectric Layer
Schematic of a simple cantilever geometry.
Natural (resonance) frequency (𝑓𝑓𝑛𝑛) and spring constant (𝑘𝑘) of cantilever are expressed below.
𝑓𝑓1 =61 𝑘𝑘𝐻𝐻𝑧𝑧
𝑓𝑓5 =1070 𝑘𝑘𝐻𝐻𝑧𝑧
𝑓𝑓6 =2090 𝑘𝑘𝐻𝐻𝑧𝑧
𝑓𝑓2 =383 𝑘𝑘𝐻𝐻𝑧𝑧
𝑓𝑓3 =421 𝑘𝑘𝐻𝐻𝑧𝑧
𝑓𝑓4 =727 𝑘𝑘𝐻𝐻𝑧𝑧
𝑛𝑛 1 2 3 4 5 6𝜆𝜆𝑛𝑛 1.875 4.694 7.855 10.996 14.137 17.279
𝑥𝑥
𝑦𝑦
𝑧𝑧
Osman SayginerNamik Kemal University, Dept. of Mechanical Engineering
www.sayginer.com
Recently, Dagdeviren et al.* have announced newenergy harvesting approach which aims togenerating energy for pacemakers. They havedeveloped a PZT cantilever based on device thatuses the motion of heartbeats (as well as themotion of the lungs and diaphragm) to power animplanted pacemaker.*Conformal piezoelectric energy harvesting and storage from motions of the heart, lung, and diaphragm." National Academy of Sciences (2014)
