Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester were studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained, and then the conditions of stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the theoretical and numerical results were proved by experiments. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields

Jia Xu

Kang-Kang Guo

Ming-Yi Luan

Zhi-Wen Zhu

English

Journal of Vibroengineering

 1240 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  
1232. Stochastic bifurcation characteristics of 
cantilevered piezoelectric energy harvester 
Jia Xu1, Ming-Yi Luan2, Zhi-Wen Zhu3, Kang-Kang Guo4 
1, 2, 3, 4School of Mechanical Engineering, Tianjin University, 92 Weijin Road, Tianjin, 300072, P. R. China 
1Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control 
92 Weijin Road, Tianjin, 300072, P. R. China 
4Corresponding author 
E-mail: 1xujia_ld@163.com, 2mingyiluan@163.com, 3zhuzhiwentju@163.com, 4guokangkangtju@163.com 
(Received 11 October 2013; received in revised form 6 December 2013; accepted 13 December 2013) 
Abstract. Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester 
were studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic 
phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric 
cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of 
the system was analyzed, and the steady-state probability density function and the joint probability 
density function of the dynamic response of the system were obtained, and then the conditions of 
stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf 
bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of 
cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the 
theoretical and numerical results were proved by experiments. The results of this paper are helpful 
to application of cantilevered piezoelectric energy harvester in engineering fields. 
Keywords: piezoelectric energy harvester, hysteretic loop, stochastic bifurcation. 
1. Introduction 
Piezoelectric ceramics is a kind of smart material. It can be used to convert mechanical energy 
into electrical energy, which is known as piezoelectric effect. Based on this effect, piezoelectric 
energy harvester can be designed to gather vibration energy of structures. Compared with other 
kinds of power generation systems, piezoelectric energy harvester has many advantages, such as 
small size, high electro-mechanical conversion efficiency, long service life, and low cost, which 
cause it be applied as green energy widely. 
Many scholars studied piezoelectric energy harvester. DuToit designed MEMS-scale 
piezoelectric mechanical vibration energy harvesters firstly [1]. Erturk developed the mechanical 
model of cantilevered piezoelectric vibration energy harvesters [2]. Priya proposed the criterion 
for material selection in design of bulk piezoelectric energy harvesters [3]. Liao studied 
parameters optimization and power characteristics of piezoelectric energy harvesters with RC 
circuit [4]. Although many advances were obtained, the modeling problem limits the application 
of cantilever piezoelectric energy harvester in industry fields. In order to optimize piezoelectric 
energy harvester effectively, it is necessary to build a model in high accuracy to describe the 
nonlinear characteristics of piezoelectric energy harvester. 
Due to the hysteretic characteristics of piezoelectric ceramics, most of the piezoelectric models 
were shown as equations with subsection function or double integral function, which were hard to 
be analyzed in theory [5-9]. Usually, research results could only be obtained by numerical or 
experiment method [10, 11]. In this paper, hysteretic nonlinear theory was introduced to develop 
a new kind of continuous piezoelectric model, and then stochastic bifurcation characteristics of 
cantilevered piezoelectric energy harvester were analyzed. 
2. Hysteretic nonlinear model of piezoelectric energy harvester 
The voltage-displacement curve of piezoelectric ceramics was shown in Fig. 1. Obviously, 
there is hysteretic nonlinearity in piezoelectric ceramics. 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1241 
 
Fig. 1. The displacement-voltage curves of piezoelectric ceramics 
In this paper, Von del Pol hysteretic model was introduced to describe the hysteretic nonlinear 
characteristics of piezoelectric ceramics. The initial Von del Pol hysteretic model describes 
hysteretic loop which is symmetrical about the initial point (0, 0). It can be shown as follows: 
 =  = 	 +  1 − 
  , (1)
where 	 is skeleton curve of hysteretic loop and usually expressed in polynomial function,  
and  are coefficients which determine the difference between the skeleton curve and the real 
curve. The essence of Von del Pol item  1 − 
   are two parabolic lines which are 
symmetrical about the original point (0, 0). 
Supposing the displacement-voltage curves of piezoelectric ceramics is symmetrical about the 
point 	, 	, it can be shown as follows: 
 − 	 =  − 	 +  − 	 +  1 −  − 	 
  , (2)
where  is voltage,  is displacement, ! (" = 1, 2, 3, 4) are coefficients, skeleton curve is chosen 
as 	 =  + .  = 	 since the loading curve have the same value as the unloading curve when  = 0, and &	 − 	 − 	 = 0 because the initial voltage of piezoelectric ceramics is zero. Thus, Eq. (2) 
can be rewritten as follows: 
 =  +  +  +  − ' , (3)
where  =  + 3	,  = −3	,  = ,  = () , ' = ()*. 
The structure of cantilevered piezoelectric energy harvester was shown in Fig. 2. It can be 
regarded as cantilever composite beam. The mechanical model of cantilevered piezoelectric 
energy harvester was shown in Fig. 3, where the thickness of adhesive layer was ignored. 
According to Hamilton's principle, we obtained: 
+ [-. − / + 01 + -0]34 = 0
5*
56
, (4)
where . is kinetic energy of structure, / is potential energy of structure, 01 is electric energy of 
piezoelectric ceramics, 0 is work done by external force: 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
1242 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  
. = 12 + 78̅: 38̅;<=
+ 12 + 7>̅: 3>̅;?@
, (5)
/ = 12 + &8̅A8̅38̅;<=
+ 12 + &>̅A>̅3>̅;?@
, (6)
01 = 12 + BC3>̅;?@
, (7)
-0 = + -:D3 +E
	
-:F, 4D + -GH=, (8)
where 7! " = I̅,  J@  is density of substrate and PZT piezoelectric ceramics, where the subscript I̅ 
is substrate, J̅ is PZT piezoelectric ceramics; ! is volume, &! is stress, A! is strain; : = :, 4 is 
deflection of composite cantilever beam, B  is electric field intensity, C  is electrostrictive 
displacement, D = KL4 is axial stochastic excitation, K is intensity of stochastic excitation, L4 
is Gauss white nosie whose mean is zero and intensity is 2C, G is electric potential, H= is electric 
charge. 
 
Fig. 2. The structure of cantilevered piezoelectric energy harvester 
 
Fig. 3. The mechanical model of cantilevered piezoelectric energy harvester 
The dynamic equation of piezoelectric cantilever beam subjected to axial stochastic excitation 
can be obtained as follows: 
M + 2NO + P − Q + QR + 12 QR − R + R = KLt, (9)
where NO is damping, P is stiffness, Q and R! " =  1, 2, 3, 4 are coefficients. 
Substituting Eq. (3) into Eq. (9), we obtained: 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1243 
M + T1 + T  − U + U + U + U + U'' + UVV = KL4, (10)
where: 
T = 2NO − Q − ' + QR − ' + QR − '       +QR − ' + QR − ', (11)
T = 12 QR − ', (12)U = P + Q, (13)
U = Q R −  + 12 R − R, (14)U = QR −  +  + R, (15)
U = Q R + 12 R + R, (16)U' = QR, (17)
UV = 12 QR. (18)
3. Stochastic stability analysis of system 
Let  = H,  = J, Eq. (10) can also be shown as follows: 
XH = J,J = −T1HJ − THJ + UH − UH − UH − UH − U'H' − UVHV + KHL4. (19)
The Hamiltonian function of Eq. (19) can be shown as follows: 
Y = 12 J −
1
2 UH +
1
3 UH +
1
4 UH +
1
5 UH' +
1
6 U'HV +
1
7 UVH]. (20)
According to the quasi-nonintegrable Hamiltonian system theory, the Hamiltonian function Y4 converges weakly in probability to an one-dimensional Ito diffusion process. The averaged 
Ito equation about the Hamiltonian function can be shown as follows: 
3Y = ^Y34 + &Y3U4, (21)
where U4 is standard Wiener process, ^Y and &Y are drift and diffusion coefficients of Ito 
stochastic process, which can be obtained in stochastic averaging method as follows: 
^Y = −NmY` − 14 QR` − 
1
8 Q' +
1
4 QR ` 
            + 116 QR' − `' + √2KCY`, 
(22)
&Y = √22 KCY`2, (23)
where ` is the solution of the following equation: 
− 12 U` +
1
3 U` +
1
4 U` +
1
5 U`' +
1
6 U'`V +
1
7 UV`] = Y. (24)
Then the associated largest Lyapunov exponent of the system is: 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
1244 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  
c = lim5→g
1
4 lnY ⁄ =
√2QR − KC4P + Q2 . (25)
Now the local stochastic stability of the system can be discussed as follows: 
1) The trivial solution Y = 0 is locally asymptotic stable if and only if c < 0, which means the 
vibration amplitude of the system will tend to zero; 
2) The trivial solution Y = 0 is locally asymptotic unstable if and only if c > 0, which means 
the vibration amplitude of the system will tend to be large; 
3) Bifurcation should appear near the trivial solution Y = 0 if and only if c = 0, which means 
the vibration amplitude of the system will jump between the small value and the large value. 
The largest Lyapunov exponent can only estimate the local stability. In this paper, the 
boundary classification method was used to analyze the global stability of the trivial solution of 
the system. Generally, the boundaries of diffusion process are singular, and the boundary 
classification is often determined by diffusion exponent, drift exponent and character value [12]. 
When Y → 0,  we obtained that:  ^Y → OY,  &Y → OY,  Rl = 2,  ml = 1,  
nl = √2o2p1q)r1* , where Rl is diffusion exponent, ml is drift exponent, nl is character value, F is left 
boundary. Thus, the left boundary Y = 0 belongs to the first kind of singular boundary. 
According to the classification for singular boundary, we obtained: 
1) The left boundary Y = 0 is repulsively natural if nl > 1; 
2) The left boundary Y = 0 is strictly natural if nl = 1; 
3) The left boundary Y = 0 is attractively natural if nl < 1. 
When Y → ∞,  we obtained that: ^Y = OY8 7⁄ ,  &Y = OY9 7⁄ ,  Rt = u] ,  mt = v] , 
where w is the right boundary. Thus, the right boundary Y = ∞ belongs to the first kind of singular 
boundary. Thus, the right boundary Y = ∞ is an entrance boundary. 
The necessary and sufficient conditions for globally asymptotic stability of the trivial solution 
require that the left undary be attractively natural and the right boundary be entrance. Thus, the 
trivial solution Y = 0  is globally asymptotically stable if and only if nl < 1, which means √2QR < KC. The influence of the character value to the stability was shown in Fig. 4. 
 
Fig. 4. The influence of the character value to the stability 
4. Stochastic bifurcation and simulation 
The averaged FPK equation of Eq. (19) is: 
x
x4 = −
x
xY [^Y] +
1
2
x[&Y]
xY , (26)
where  is probability density. 
Thus, the stationary probability density function of the system is: 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1245 
Y = T̅Yyexp 2√2CK } √2  Nm 
18 Q'  14 QR Y~ 14 QRY~  116 QR'  Y ⁄ , (27)
where T̅ is a normalization constant,   √2o2p1q)~r1*r1* . 
The result of numerical simulation were shown in Fig. 5-8, where P  0.5, C  0.5, F  1, O  0.05,   40, B  2×1011, T  8×10-4,   6×10-11,   3.8,   –0.27,   0.014. 
 
 
Fig. 5. The steady-state probability density of the 
system when   2 and '  4.5 Fig. 6. The joint probability density of the system when   2 and '  4.5 
 
 
Fig. 7. The steady-state probability density of the 
system when   3.5 and '  4 Fig. 8. The joint probability density of the system  when   3.5 and '  4 
From Fig. 5-8, we can see that: 
1) The hysteretic nonlinear damping coefficients !  ("  4, 5) can induce stochastic Hopf 
bifurcation of the system. From Fig. 6 and Fig. 8, we can obviously see that there are two limit 
cycles in the stationary probability density, which means that there are two vibration amplitudes 
whose probability are both very high. Jumping phenomena between the two vibration amplitudes 
will appear when the conditions are changed; 
2) The stationary probability density of the response of the system can be changed through 
adjusting the parameters ! ("  4, 5). It means that different PZT piezoelectric ceramics materials 
will cause different vibration amplitudes of the system since the parameters !  ("  4, 5) are 
5 10 15 20
H
0.05
0.10
0.15
0.20
0.25
0.30
p
5 10 15 20
H
0.05
0.10
0.15
p
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
1246 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  
determined by PZT piezoelectric ceramics. It provide a way to improve the efficiency of the 
energy harvester since the stationary probability density of the big vibration amplitude can be 
increased by choosing appropriate PZT piezoelectric ceramics. 
The experimental results of cantilevered piezoelectric energy harvester were shown in Fig. 9 
and Fig. 10, where the vibration amplitude was shown as output voltage of sensor. We can see 
that system paremeters !  (" = 4, 5) can influence the vibration of the system, and jumping 
phenomena between the two vibration amplitudes appears when the conditions are changed. 
  
Fig. 9. The response of cantilevered piezoelectric energy harvester when  = 3.5 and ' = 4 
 
Fig. 10. The response of cantilevered piezoelectric energy harvester when  = 2 and ' = 4.5 
5. Conclusions 
Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester had been 
studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic 
phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric 
cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of 
the system was analyzed, and the steady-state probability density function and the joint probability 
density function of the dynamic response of the system were obtained, and then the conditions of 
stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf 
bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of 
cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the 
theoretical and numerical results were proved by experiments. The results of this paper are helpful 
to application of cantilevered piezoelectric energy harvester in engineering fields. 
Acknowledgements 
The authors gratefully acknowledge the support of Natural Science Foundation of China 
(NSFC) through grant No. 11272229 and 11302144, the Ph. D. Programs Foundation of Ministry 
of Education of China through grant No. 20120032120006, and Tianjin Research Program of 
1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  
JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1247 
Application Foundation and Advanced Technology through grant No. 13JCYBJC17900. 
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International Journal of Non-Linear Mechanics, Vol. 34, Issue 3, 1999, p. 437-447. 


Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester

Xu, Jia

Luan, Ming-Yi

Zhu, Zhi-Wen

Guo, Kang-Kang

JVE International

 1240 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  1232. Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester Jia Xu1, Ming-Yi Luan2, Zhi-Wen Zhu3, Kang-Kang Guo4 1, 2, 3, 4School of Mechanical Engineering, Tianjin University, 92 Weijin Road, Tianjin, 300072, P. R. China 1Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control 92 Weijin Road, Tianjin, 300072, P. R. China 4Corresponding author E-mail: 1xujia_ld@163.com, 2mingyiluan@163.com, 3zhuzhiwentju@163.com, 4guokangkangtju@163.com (Received 11 October 2013; received in revised form 6 December 2013; accepted 13 December 2013) Abstract. Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester were studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained, and then the conditions of stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the theoretical and numerical results were proved by experiments. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields. Keywords: piezoelectric energy harvester, hysteretic loop, stochastic bifurcation. 1. Introduction Piezoelectric ceramics is a kind of smart material. It can be used to convert mechanical energy into electrical energy, which is known as piezoelectric effect. Based on this effect, piezoelectric energy harvester can be designed to gather vibration energy of structures. Compared with other kinds of power generation systems, piezoelectric energy harvester has many advantages, such as small size, high electro-mechanical conversion efficiency, long service life, and low cost, which cause it be applied as green energy widely. Many scholars studied piezoelectric energy harvester. DuToit designed MEMS-scale piezoelectric mechanical vibration energy harvesters firstly [1]. Erturk developed the mechanical model of cantilevered piezoelectric vibration energy harvesters [2]. Priya proposed the criterion for material selection in design of bulk piezoelectric energy harvesters [3]. Liao studied parameters optimization and power characteristics of piezoelectric energy harvesters with RC circuit [4]. Although many advances were obtained, the modeling problem limits the application of cantilever piezoelectric energy harvester in industry fields. In order to optimize piezoelectric energy harvester effectively, it is necessary to build a model in high accuracy to describe the nonlinear characteristics of piezoelectric energy harvester. Due to the hysteretic characteristics of piezoelectric ceramics, most of the piezoelectric models were shown as equations with subsection function or double integral function, which were hard to be analyzed in theory [5-9]. Usually, research results could only be obtained by numerical or experiment method [10, 11]. In this paper, hysteretic nonlinear theory was introduced to develop a new kind of continuous piezoelectric model, and then stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester were analyzed. 2. Hysteretic nonlinear model of piezoelectric energy harvester The voltage-displacement curve of piezoelectric ceramics was shown in Fig. 1. Obviously, there is hysteretic nonlinearity in piezoelectric ceramics. 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1241  Fig. 1. The displacement-voltage curves of piezoelectric ceramics In this paper, Von del Pol hysteretic model was introduced to describe the hysteretic nonlinear characteristics of piezoelectric ceramics. The initial Von del Pol hysteretic model describes hysteretic loop which is symmetrical about the initial point (0, 0). It can be shown as follows:  =  = 	 +  1 −   , (1)where 	 is skeleton curve of hysteretic loop and usually expressed in polynomial function,  and  are coefficients which determine the difference between the skeleton curve and the real curve. The essence of Von del Pol item  1 −    are two parabolic lines which are symmetrical about the original point (0, 0). Supposing the displacement-voltage curves of piezoelectric ceramics is symmetrical about the point 	, 	, it can be shown as follows:  − 	 =  − 	 +  − 	 +  1 −  − 	   , (2)where  is voltage,  is displacement, ! (" = 1, 2, 3, 4) are coefficients, skeleton curve is chosen as 	 =  + .  = 	 since the loading curve have the same value as the unloading curve when  = 0, and &	 − 	 − 	 = 0 because the initial voltage of piezoelectric ceramics is zero. Thus, Eq. (2) can be rewritten as follows:  =  +  +  +  − ' , (3)where  =  + 3	,  = −3	,  = ,  = () , ' = ()*. The structure of cantilevered piezoelectric energy harvester was shown in Fig. 2. It can be regarded as cantilever composite beam. The mechanical model of cantilevered piezoelectric energy harvester was shown in Fig. 3, where the thickness of adhesive layer was ignored. According to Hamilton's principle, we obtained: + [-. − / + 01 + -0]34 = 05*56, (4)where . is kinetic energy of structure, / is potential energy of structure, 01 is electric energy of piezoelectric ceramics, 0 is work done by external force: 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 1242 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  . = 12 + 78̅: 38̅;<=+ 12 + 7>̅: 3>̅;?@, (5)/ = 12 + &8̅A8̅38̅;<=+ 12 + &>̅A>̅3>̅;?@, (6)01 = 12 + BC3>̅;?@, (7)-0 = + -:D3 +E	-:F, 4D + -GH=, (8)where 7! " = I̅,  J@  is density of substrate and PZT piezoelectric ceramics, where the subscript I̅ is substrate, J̅ is PZT piezoelectric ceramics; ! is volume, &! is stress, A! is strain; : = :, 4 is deflection of composite cantilever beam, B  is electric field intensity, C  is electrostrictive displacement, D = KL4 is axial stochastic excitation, K is intensity of stochastic excitation, L4 is Gauss white nosie whose mean is zero and intensity is 2C, G is electric potential, H= is electric charge.  Fig. 2. The structure of cantilevered piezoelectric energy harvester  Fig. 3. The mechanical model of cantilevered piezoelectric energy harvester The dynamic equation of piezoelectric cantilever beam subjected to axial stochastic excitation can be obtained as follows: M + 2NO + P − Q + QR + 12 QR − R + R = KLt, (9)where NO is damping, P is stiffness, Q and R! " =  1, 2, 3, 4 are coefficients. Substituting Eq. (3) into Eq. (9), we obtained: 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1243 M + T1 + T  − U + U + U + U + U'' + UVV = KL4, (10)where: T = 2NO − Q − ' + QR − ' + QR − '       +QR − ' + QR − ', (11)T = 12 QR − ', (12)U = P + Q, (13)U = Q R −  + 12 R − R, (14)U = QR −  +  + R, (15)U = Q R + 12 R + R, (16)U' = QR, (17)UV = 12 QR. (18)3. Stochastic stability analysis of system Let  = H,  = J, Eq. (10) can also be shown as follows: XH = J,J = −T1HJ − THJ + UH − UH − UH − UH − U'H' − UVHV + KHL4. (19)The Hamiltonian function of Eq. (19) can be shown as follows: Y = 12 J −12 UH +13 UH +14 UH +15 UH' +16 U'HV +17 UVH]. (20)According to the quasi-nonintegrable Hamiltonian system theory, the Hamiltonian function Y4 converges weakly in probability to an one-dimensional Ito diffusion process. The averaged Ito equation about the Hamiltonian function can be shown as follows: 3Y = ^Y34 + &Y3U4, (21)where U4 is standard Wiener process, ^Y and &Y are drift and diffusion coefficients of Ito stochastic process, which can be obtained in stochastic averaging method as follows: ^Y = −NmY` − 14 QR` − 18 Q' +14 QR `             + 116 QR' − `' + √2KCY`, (22)&Y = √22 KCY`2, (23)where ` is the solution of the following equation: − 12 U` +13 U` +14 U` +15 U`' +16 U'`V +17 UV`] = Y. (24)Then the associated largest Lyapunov exponent of the system is: 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 1244 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  c = lim5→g14 lnY ⁄ =√2QR − KC4P + Q2 . (25)Now the local stochastic stability of the system can be discussed as follows: 1) The trivial solution Y = 0 is locally asymptotic stable if and only if c < 0, which means the vibration amplitude of the system will tend to zero; 2) The trivial solution Y = 0 is locally asymptotic unstable if and only if c > 0, which means the vibration amplitude of the system will tend to be large; 3) Bifurcation should appear near the trivial solution Y = 0 if and only if c = 0, which means the vibration amplitude of the system will jump between the small value and the large value. The largest Lyapunov exponent can only estimate the local stability. In this paper, the boundary classification method was used to analyze the global stability of the trivial solution of the system. Generally, the boundaries of diffusion process are singular, and the boundary classification is often determined by diffusion exponent, drift exponent and character value [12]. When Y → 0,  we obtained that:  ^Y → OY,  &Y → OY,  Rl = 2,  ml = 1,  nl = √2o2p1q)r1* , where Rl is diffusion exponent, ml is drift exponent, nl is character value, F is left boundary. Thus, the left boundary Y = 0 belongs to the first kind of singular boundary. According to the classification for singular boundary, we obtained: 1) The left boundary Y = 0 is repulsively natural if nl > 1; 2) The left boundary Y = 0 is strictly natural if nl = 1; 3) The left boundary Y = 0 is attractively natural if nl < 1. When Y → ∞,  we obtained that: ^Y = OY8 7⁄ ,  &Y = OY9 7⁄ ,  Rt = u] ,  mt = v] , where w is the right boundary. Thus, the right boundary Y = ∞ belongs to the first kind of singular boundary. Thus, the right boundary Y = ∞ is an entrance boundary. The necessary and sufficient conditions for globally asymptotic stability of the trivial solution require that the left undary be attractively natural and the right boundary be entrance. Thus, the trivial solution Y = 0  is globally asymptotically stable if and only if nl < 1, which means √2QR < KC. The influence of the character value to the stability was shown in Fig. 4.  Fig. 4. The influence of the character value to the stability 4. Stochastic bifurcation and simulation The averaged FPK equation of Eq. (19) is: xx4 = −xxY [^Y] +12x[&Y]xY , (26)where  is probability density. Thus, the stationary probability density function of the system is: 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1245 Y = T̅Yyexp 2√2CK } √2  Nm 18 Q'  14 QR Y~ 14 QRY~  116 QR'  Y ⁄ , (27)where T̅ is a normalization constant,   √2o2p1q)~r1*r1* . The result of numerical simulation were shown in Fig. 5-8, where P  0.5, C  0.5, F  1, O  0.05,   40, B  2×1011, T  8×10-4,   6×10-11,   3.8,   –0.27,   0.014.   Fig. 5. The steady-state probability density of the system when   2 and '  4.5 Fig. 6. The joint probability density of the system when   2 and '  4.5   Fig. 7. The steady-state probability density of the system when   3.5 and '  4 Fig. 8. The joint probability density of the system  when   3.5 and '  4 From Fig. 5-8, we can see that: 1) The hysteretic nonlinear damping coefficients !  ("  4, 5) can induce stochastic Hopf bifurcation of the system. From Fig. 6 and Fig. 8, we can obviously see that there are two limit cycles in the stationary probability density, which means that there are two vibration amplitudes whose probability are both very high. Jumping phenomena between the two vibration amplitudes will appear when the conditions are changed; 2) The stationary probability density of the response of the system can be changed through adjusting the parameters ! ("  4, 5). It means that different PZT piezoelectric ceramics materials will cause different vibration amplitudes of the system since the parameters !  ("  4, 5) are 5 10 15 20H0.050.100.150.200.250.30p5 10 15 20H0.050.100.15p1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO 1246 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716  determined by PZT piezoelectric ceramics. It provide a way to improve the efficiency of the energy harvester since the stationary probability density of the big vibration amplitude can be increased by choosing appropriate PZT piezoelectric ceramics. The experimental results of cantilevered piezoelectric energy harvester were shown in Fig. 9 and Fig. 10, where the vibration amplitude was shown as output voltage of sensor. We can see that system paremeters !  (" = 4, 5) can influence the vibration of the system, and jumping phenomena between the two vibration amplitudes appears when the conditions are changed.   Fig. 9. The response of cantilevered piezoelectric energy harvester when  = 3.5 and ' = 4  Fig. 10. The response of cantilevered piezoelectric energy harvester when  = 2 and ' = 4.5 5. Conclusions Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester had been studied in this paper. Von de Pol differencial item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained, and then the conditions of stochastic Hopf bifurcation were analyzed. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. Finally, the theoretical and numerical results were proved by experiments. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields. Acknowledgements The authors gratefully acknowledge the support of Natural Science Foundation of China (NSFC) through grant No. 11272229 and 11302144, the Ph. D. Programs Foundation of Ministry of Education of China through grant No. 20120032120006, and Tianjin Research Program of 1232. STOCHASTIC BIFURCATION CHARACTERISTICS OF CANTILEVERED PIEZOELECTRIC ENERGY HARVESTER.  JIA XU, MING-YI LUAN, ZHI-WEN ZHU, KANG-KANG GUO  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY 2014. VOLUME 16, ISSUE 3. ISSN 1392-8716 1247 Application Foundation and Advanced Technology through grant No. 13JCYBJC17900. References [1] DuToit N. E., Wardle B. L., Kim S. G. Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters. Integrated Ferroelectrics, Vol. 71, Issue 1, 2005, p. 121-160. [2] Erturk S. A., Inman D. J. On mechanical modeling of cantilevered piezoelectric vibration energy harvesters. Journal of Intelligent Material Systems and Structures, Vol. 19, Issue 11, 2008, p. 1311-1325. [3] Priya P., Shashank M. Criterion for material selection in design of bulk piezoelectric energy harvesters. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, Vol. 57, Issue 12, 2010, p. 2610-2612. [4] Liao M. R. Y. B., Sodano H. A. 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Stochastic bifurcation characteristics of cantilevered piezoelectric energy harvester

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