This document summarizes a report analyzing a multi-directional nonlinear piezoelectric vibration energy harvester. It begins with an introduction discussing challenges with existing energy harvesters and previous work addressing those challenges. It then outlines the derivation of formulas for a proposed tri-direction dual-beam piezoelectric vibration energy harvester (TDPVEH) model. This includes models of a piezoelectric cantilever beam, magnetic charges, and the electromechanical coupling of the system. The document concludes by introducing the TDPVEH prototype combining a beam-spring-beam structure to enable multi-directional energy harvesting.

1. Analysis and Realization of Multi-Direction Nonlinear
Piezoelectric Vibration Energy Harvester
July 2017
Reporter : Hsuan-Chen Lu
Adviser : Wei-Jiun Su
2018/1/22 1

2. Outline
• Introduction
• Formula Derivation
• Analysis of System Potential Energy
• Prototype and experiment set-up
• Experimental Verification and System Performance
• Conclusions and Future Work
2018/1/22 2

3. Introduction
2018/1/22 3
Wireless Sensor NodesIoT Energy Harvester

4. Introduction
• electromagnetic transducer • electrostatic transducer
2018/1/22 4
• cantilever piezoelectric energy
harvester
S
N
coil
Vibration Energy Harvester

5. Introduction
• Challenge 1: Narrow resonance
bandwidth
• Challenge 2: Single operating
direction
2018/1/22 5

6. Introduction
• To solve the problem of narrow resonance bandwidth
2018/1/22 6
Meandering PVEH[19] Passive-Self-Tunable PVEH[24]

7. Introduction
• To solve the problem of narrow resonance bandwidth
2018/1/22 7
PVEH array[27] Magnetic PVEH[28]

8. Introduction
• To solve the problem of single operating direction
2018/1/22 8
Dandelion-like multi-directional PVEH[33]Miniature Three-Axis PVEH[34] Spiral-shaped PVEH[36]

9. Introduction
• Tri-direction Dual-beam PVEH
2018/1/22 9
Auxiliary Beam
Main Beam
PZT
Spring-mass System
Magnet
X(1)
Y(2)
Z(3)

10. Formula
Derivation
2018/1/22 10
1. Piezoelectric cantilever beam model
2. Magnetic charge model
3. TDPVEH
4. Implement in Matlab

11. Piezoelectric Cantilever Beam Model
2018/1/22 11
22 2
2 2 2
(t)(x ,t) (x ,t) (x ,t) bI I rI I rI I
a I I
I
d wM w w
c m m
x t t dt
  
   
  
 
22 2
2 2 2
(t)(x ,t) (x ,t) (x ,t)
(x L ) bII II rII II rII II
a II II t II II
II
d wM w w
c m m M
x t t dt

  
     
  

12. Piezoelectric Cantilever Beam Model
• Part 1 (PZT composite beam) • Part 2 (Substrate beam only)
2018/1/22 12
𝑏 𝑝
𝑏𝑠
ℎ 𝑎
ℎ 𝑐
ℎ 𝑏
ℎ 𝑝
ℎ 𝑠
Y(2)
Z(3)
𝑏𝑠
ℎ 𝑠
Y(2)
Z(3)
1 1(x ,t)
b c
a b
h h
s p
I I s p
h h
M T b zdz T b zdz   
2
1
2
(x ,t)
s
s
h
s
II II s
h
M T b zdz

 

13. Piezoelectric Cantilever Beam Model
• For Substrate (Hooke’s Law)
• For PZT ceramic (Piezoelectric constitutive relation (d-form))
2018/1/22 13
1 1 31 3
1
S d Ep p
p
T
Y
 
2
1 1 31 3 1 32
(x ,t) (t)
(S d E ) ,Ep p p rk k
p
k p
w v
T Y S z
x h
 
       
2
1 1 1 2
(x ,t)s s s rk k
s
k
w
T Y S S z
x
 
  
 

14. Piezoelectric Cantilever Beam Model
2018/1/22 14
 
24 2
4 2 2
(t)(x ,t) (x ,t) (x ,t)
(t) (x ) (x ) brI I rI I rI I
I a I v I I I I
I
d ww w w
YI c m Q v H H L m
x t t dt
  
      
  
 
24 2
4 2 2
(t)(x ,t) (x ,t) (x ,t)
(x L ) brII II rII II rII II
II a II II t II II
II
d ww w w
YI c m m M
x t t dt

  
     
  

15. Piezoelectric Cantilever Beam Model
• Boundary conditions and continuous conditions
2018/1/22 15
(0) 0Ii 
(0)
0Ii
Ix



2
2
2
(L )
(L ) 0IIi II
II i t IIi II
II
d
YI M
dx

  
2
2
2
(L ) (L )
0IIi II IIi II
II i t
II II
d d
YI J
dx dx
 
 
(L ) (0)Ii I IIi 
(L ) (0)Ii I IIi
I II
d d
dx dx
 

2 2
2 2
(L ) (0)Ii I IIi
I II
I II
d d
YI YI
dx dx
 

3 3
3 3
(L ) (0)Ii I IIi
I II
I II
d d
YI YI
dx dx
 


16. Piezoelectric Cantilever Beam Model
2018/1/22 16
6 2 6 2 6 2
0
0
0
0
0
0
AB EF GH
A
B
E
M M M
F
G
H
  
   
   
   
   
      
   
   
   
   

17. Piezoelectric Cantilever Beam Model
2018/1/22 17
 
24 2
4 2 2
(t)(x ,t) (x ,t) (x ,t)
(t) (x ) (x ) brI I rI I rI I
I a I v I I I I
I
d ww w w
YI c m Q v H H L m
x t t dt
  
      
  
 
24 2
4 2 2
(t)(x ,t) (x ,t) (x ,t)
(x L ) brII II rII II rII II
II a II II t II II
II
d ww w w
YI c m m M
x t t dt

  
     
  
2
(t) 2 (t) (t) (t) F (t)i i i i i i i iv         && &
1
(x ,t) (x ) (t)rI I Ii I i
i
w 


 
1
(x ,t) (x ) (t)rII II IIi II i
i
w 


 

18. Piezoelectric Cantilever Beam Model
• Piezoelectric constitutive relation (e-form)
• Gauss’s Law and Ohm’s Law
2018/1/22 18
2
3 31 332
(x ,t) (t)
(x ,t) SIi I
I p pc
I p
w v
D d Y h
x h


  

  (t)
(t)
l
d v
D ndA i
dt R
  
v v 1
(t) (t) (t)p i i
l
C v v
R
   &&

19. Piezoelectric Cantilever Beam Model
• Electromechanical model
2018/1/22 19
2
(t) 2 (t) (t) (t) F (t)
1
(t) (t) (t)
i i i i i i i i
p i i
l
v
C v v
R
         
   
&& &
&&

20. Magnetic Charge Model
• Charge model
2018/1/22 20
ˆi iM n  
   
1/22 2 2
d x X y Y         
 
1 2
4 o
u
d
 


1 1 1 1 1 1
1 2
, , , ,
0 0 0 0 0 0
( 1) (p ,q ,s ,r)
4
i j k l m n
x y z x y z ij kl mn
i j k l m no
M M
F 

    
     
 
2 2 2 2
2 22 2
a b A B
a b A B
U udYdXdydx
   
    
F U 
r

21. Introduction of TD-VEH
2
(t) 2 (t) (t) (t) (t)
1
(t) (t) (t)
i i i i i i i i
p i i
l
v F
C v v
R
         
   
&& &
&&
1 1 1 1 1 1
1 2
, , , ,
0 0 0 0 0 0
( 1) (p ,q ,s ,r)
4
i j k l m n
x y z x y z ij kl mn
i j k l m no
M M
F 

    
     
 
2018/1/22 21
Piezoelectric Cantilevered
Beam Model
Charge Model

22. Introduction of TD-VEH
2
(t) 2 (t) (t) (t) (t) (t)
1(t) (t) (t)
(t) (t) W (t) (t) (t)
m m m m m m m m sz mb
p m m
l
s s s s s s sx sb
v F F
C v v
R
M W C W K F F
           
   
   
&& &
&&
&& &
2018/1/22 22
• Beam-Spring

23. Introduction of TD-VEH
2
2
(t) 2 (t) (t) (t) (t) (t)
1(t) (t) (t)
(t) 2 (t) (t) (t) (t)
m m m m m m m m az mb
p m m
l
a a a a a a a ay ab
v F F
C v v
R
F F
  
  
         
   
       
&& &
&&
&& &
2018/1/22 23
• Beam-Beam

24. Tri-direction Dual-beam VEH
2018/1/22 24
2
2
(t) 2 (t) (t) ( (t) (t)) (t) (t)
(t) 2 (t) (t) (t) (t)
(t) (t) W (t) (t) (t)
1(t) (t) (t)
m m m m m m m az sz m mb
a a a a a a a ay ab
s s s s s s sx sb
p m m
l
F F v F
F F
M W C W K F F
C v v
R
  
  
          
       
   
   
&& &
&& &
&& &
&&
(Main Beam)
(Auxiliary Beam)
(Spring-mass System)
(Circuit)

25. Implement in Matlab
• Architecture
2018/1/22 25
main.m
Mainbeam.m
Auxiliarybeam.m
Springmass.m
Magforce_cube.m
Response.m
2
2
(t) 2 (t) (t) ( (t) (t)) (t) (t)
(t) 2 (t) (t) (t) (t)
(t) (t) W (t) (t) (t)
1(t) (t) (t)
m m m m m m m az sz m mb
a a a a a a a ay ab
s s s s s s sx sb
p m m
l
F F v F
F F
M W C W K F F
C v v
R
  
  
          
       
   
   
&& &
&& &
&& &
&&

26. Analysis of
System Potential
Energy
2018/1/22 26
1. Beam-spring
• Straight configuration
• Dislocated configuration
2. Beam-beam
• Straight configuration
• Dislocated configuration

27. Analysis of System Potential Energy
• Beam-Spring
𝑊𝑡𝑜𝑡𝑎𝑙=
2018/1/22 27
𝑊𝑠𝑚 𝑊 𝑚𝑏 𝑊𝑚𝑎𝑔+ +

28. Analysis of System Potential Energy
• Beam-Spring
𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑊𝑠𝑚 + 𝑊 𝑚𝑏 + 𝑊𝑚𝑎𝑔
2018/1/22 28
X(1)
Y(2)
Z(3)

29. Analysis of System Potential Energy
• Beam-Beam
𝑊𝑡𝑜𝑡𝑎𝑙=
2018/1/22 29
𝑊𝑎𝑏 𝑊 𝑚𝑏 𝑊𝑚𝑎𝑔+ +

30. Analysis of System Potential Energy
• Beam-Beam
𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑊𝑎𝑏 + 𝑊 𝑚𝑏 + 𝑊𝑚𝑎𝑔
2018/1/22 30
X(1)
Y(2)
Z(3)
8mm7mm6mm

31. Analysis of System Potential Energy
• Beam-Beam
𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑊𝑎𝑏 + 𝑊 𝑚𝑏 + 𝑊𝑚𝑎𝑔
2018/1/22 31
X(1)
Y(2)
Z(3)
6mm
1.5mm
1.5mm

32. Prototype and
Experiment Set-
up
2018/1/22 32
1. Prototype design
2. Experiment set-up

33. Prototype Design
• Main beam • Auxiliary beam
2018/1/22 33
• Spring mass

34. Prototype Design
• Z(3) excitation • Y(2) excitation
2018/1/22 34
• X(1) excitation

35. Experiment Set-up
2018/1/22 35

36. Experimental
Verification and
Result
2018/1/22 36
1. Linear PVEH
2. Beam-spring
• Z(3) excitation
• X(1) excitation
3. Beam-beam
• Z(3) excitation
• Y(2) excitation
4. Tri-direction Dual-beam PVEH
• Experimental verification
• Performance comparison

37. Linear PVEH
• Linear PVEH
2018/1/22 37
0.0145 
33 / 2710T
o  20pC nF

38. Linear PVEH
• Linear PVEH(0.2g)
2018/1/22 38
1 1
31 230 ( )d pm V pC N 
   

39. Linear PVEH
• Linear PVEH(0.2g)
2018/1/22 39

40. Beam-spring (Z(3) excitation)
• Straight configuration • Dislocated configuration(Y bias)
2018/1/22 40
• Dislocated configuration(Z bias)

41. Beam-spring Straight configuration
(Z(3) excitation)
• 𝑑 𝑠1 = 5𝑚𝑚 • 𝑑 𝑠1 = 6𝑚𝑚
2018/1/22 41
• 𝑑 𝑠1 = 7𝑚𝑚

42. Beam-spring Dislocated configuration (𝑑 𝑠3= 1.5𝑚𝑚)
(Z(3) excitation)
• 𝑑 𝑠1 = 5𝑚𝑚 • 𝑑 𝑠1 = 6𝑚𝑚
2018/1/22 42
• 𝑑 𝑠1 = 7𝑚𝑚

43. Beam-spring (X(1) excitation)
• Straight configuration • Dislocated configuration(Z bias)
2018/1/22 43

44. Beam-spring Straight configuration
(X(1) excitation)
• 𝑑 𝑠1 = 7𝑚𝑚 • 𝑑 𝑠1 = 8𝑚𝑚
2018/1/22 44
• 𝑑 𝑠1 = 9𝑚𝑚

45. Beam-spring Dislocated configuration (𝑑 𝑠3= 1.5𝑚𝑚)
(X(1) excitation)
• 𝑑 𝑠1 = 7𝑚𝑚 • 𝑑 𝑠1 = 8𝑚𝑚
2018/1/22 45
• 𝑑 𝑠1 = 9𝑚𝑚

46. Beam-beam (Z(3) excitation)
• Straight configuration • Dislocated configuration(Y bias)
2018/1/22 46

47. Beam-beam (Z(3) excitation)
• Dislocated configuration(Z bias) • Dislocated configuration(Z bias)
2018/1/22 47

48. Beam-beam Simulation of Tip Displacement
(Z(3) excitation)
• Straight • Dislocated (𝑑 𝑎3= 1.5𝑚𝑚)
2018/1/22 48
• Dislocated (𝑑 𝑎3= 3𝑚𝑚)

49. Beam-beam Straight configuration
(Z(3) excitation)
• 𝑑 𝑎1 = 5𝑚𝑚 • 𝑑 𝑎1 = 6𝑚𝑚
2018/1/22 49
• 𝑑 𝑎1 = 7𝑚𝑚

50. Beam-beam Dislocated configuration (𝑑 𝑎3= 1.5𝑚𝑚)
(Z(3) excitation)
• 𝑑 𝑎1 = 5𝑚𝑚 • 𝑑 𝑎1 = 6𝑚𝑚
2018/1/22 50
• 𝑑 𝑎1 = 7𝑚𝑚

51. Beam-beam Dislocated configuration (𝑑 𝑎3= 3𝑚𝑚)
(Z(3) excitation)
• 𝑑 𝑎1 = 5𝑚𝑚 • 𝑑 𝑎1 = 6𝑚𝑚
2018/1/22 51
• 𝑑 𝑎1 = 7𝑚𝑚

52. Beam-beam (Y(2) excitation)
• Straight configuration • Dislocated configuration(Z bias)
2018/1/22 52

53. Beam-beam Straight configuration
(Y(2) excitation)
• 𝑑 𝑎1 = 5𝑚𝑚 • 𝑑 𝑎1 = 6𝑚𝑚
2018/1/22 53
• 𝑑 𝑎1 = 7𝑚𝑚

54. Beam-beam Dislocated configuration (𝑑 𝑎3= 1.5𝑚𝑚)
(Y(2) excitation)
• 𝑑 𝑎1 = 5𝑚𝑚 • 𝑑 𝑎1 = 6𝑚𝑚
2018/1/22 54
• 𝑑 𝑎1 = 7𝑚𝑚

55. Tri-direction Dual-beam PVEH
• Straight configuration • Dislocated configuration
2018/1/22 55

56. TDPVEH Straight configuration
• Z(3) excitation • Y(2) excitation
2018/1/22 56
• X(1) excitation

57. TDPVEH Dislocated configuration
• Z(3) excitation • Y(2) excitation
2018/1/22 57
• X(1) excitation

58. Performance comparison (Z(3) excitation)
• Conventional • Straight
2018/1/22 58
• Dislocated

59. Performance comparison (Y(2) excitation)
• Straight configuration • Dislocated configuration
2018/1/22 59

60. Performance comparison (X(1) excitation)
• Straight configuration • Dislocated configuration
2018/1/22 60

61. Performance comparison
• Peak power and bandwidth graph
2018/1/22 61

62. Conclusion and Future Work
Conclusion:
• Experiments supported theory model
• Frequency response changed with different configurations
• Both two configurations improved their adaptability
2018/1/22 62

63. Conclusion and Future Work
Future Work :
• Optimization of the system
• Applications for real world
• Design of Interface circuit
• Energy harvesting from auxiliary beam
2018/1/22 63

64. 2018/1/22 64
QAQ?
(Question And Questions?)