12432


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 22, No 2, 2024, pp. 257 - 273  

https://doi.org/10.22190/FUME240118033M 

© 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

NONLINEAR DYNAMICAL CHARACTERISTICS OF HYBRID 

TRI-STABLE PIEZOELECTRIC ENERGY HARVESTER BASED 

ON ROTATIONAL MOTION 

Da-Wei Man1,2, Yu Zhang1, Li-Ping Tang1,2, Qing-Hu Xu1,2, Dong Chen1,2, 

Bang-Dong Jiang1, Ting-Ting Han1,2, Tao Xu3, Jin-Bao Li4 

1School of Civil Engineering, Anhui Jianzhu University, Hefei, China 
2BIM Engineering Center of Anhui Province, Hefei, China 

3China Construction First Group the Fifth Construction CO., Ltd, Beijing, China 
4Jiangsu Southeast Special Engineering & Technology Co., Ltd, Nanjing, China 

ORCID iDs: Da-Wei Man   https://orcid.org/0000-0003-3053-2318  
 Yu Zhang   https://orcid.org/0009-0006-4473-9183    
 Li-Ping Tang   https://orcid.org/0009-0006-5184-4483    
 Qing-Hu Xu   https://orcid.org/0000-0002-8617-8373    
 Dong Chen   https://orcid.org/0009-0002-0221-9383    
 Bang-Dong Jiang   https://orcid.org/0009-0003-0063-138X    
 Ting-Ting Han   https://orcid.org/0009-0008-2379-214X    
 Tao Xu   https://orcid.org/0009-0002-8384-1608    
 Jin-Bao Li   https://orcid.org/0009-0009-0251-0665    

Abstract. The paper presents an improved hybrid tri-stable cantilever piezoelectric energy 
harvester based on rotational motion, thus providing a new perspective for achieving higher 
efficiency in energy capture for rotational motion. The proposed system comprises a 
piezoelectric cantilever beam with an innovative dynamic amplifier installed at the edge of a 
vehicle's wheel hub. Through theoretical analysis and numerical simulations, the influence of 
parameters such as the mass of the tip magnetic on the piezoelectric beam, the wheel hub 
radius, rotational speed, and the ratio of the dynamic amplifier's spring stiffness are 
investigated with respect to the system's steady-state dynamic response and time-domain 
performance. A comparative analysis is also conducted with traditional tri-stable piezoelectric 
energy harvesters. The results demonstrate that the hybrid tri-stable piezoelectric energy 
harvester exhibits superior performance in capturing vibrational energy during rotational 
motion, compared to traditional tri-stable piezoelectric energy harvesters. Proper adjustments 
to the mass of the tip magnetic and the internal spring stiffness of the dynamic amplifier can 
enhance the system's output voltage and the rotational speed range of inter-well motion. 
Additionally, the rotational speed range of inter-well motion increases with the expansion of 
the wheel hub radius. However, when the rotational speed is below 100 rpm, the influence of 
varying the wheel hub radius on the system's output voltage is minimal. 

Key words: Rotational motion, Piezoelectric energy harvester, Tri-stable, Dynamic 

amplifier, Steady-state dynamic response 

 
Received: January 18, 2024 / Accepted May 22, 2024 

Corresponding author: Da-Wei Man 
School of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China 

E-mail: daweiman@hotmail.com 

https://orcid.org/0000-0003-3053-2318
https://orcid.org/0009-0006-4473-9183
https://orcid.org/0009-0006-5184-4483
https://orcid.org/0000-0002-8617-8373
https://orcid.org/0009-0002-0221-9383
https://orcid.org/0009-0003-0063-138X
https://orcid.org/0009-0008-2379-214X
https://orcid.org/0009-0002-8384-1608
https://orcid.org/0009-0009-0251-0665


258 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

1. INTRODUCTION 

The widespread application of wireless sensor networks in infrastructure monitoring, 

ecological environment monitoring, and disaster warning imposes higher requirements on 

the cleanliness and sustainability of their power sources. Traditional chemical batteries are 

toxic, have limited lifespans, and pose challenges such as time-consuming replacements, 

high costs, and difficulty in handling waste batteries [1-6]. Piezoelectric energy harvesters, 

capable of capturing vibrational energy from the surrounding environment and converting 

it into persistent and clean electrical energy, hold significant potential for self-powering 

wireless sensor nodes due to their simple, compact, and easily integrable structures. This 

has garnered extensive attention from scholars both domestically and internationally in 

recent years [7-12]. Early linear piezoelectric harvesters were restricted to generating high 

output power only in resonance, exhibiting low efficiency in wide-frequency or random 

vibrational environments. To overcome this limitation, researchers introduced nonlinearity 

to linear systems, resulting in nonlinear piezoelectric harvesters with broader operating 

frequency bands. This innovation renders them less sensitive to variations in external 

vibration frequencies, so that they are more suitable for effectively harvesting energy from 

diverse environmental vibrations [13, 14]. 

Stanton et al. [15, 16] established a magnetic non-linear bi-stable piezoelectric 

harvester composed of a permanent magnet and a piezoelectric cantilever beam. Under 

low-frequency vibrational excitations, it exhibits oscillations between two potential wells, 

significantly broadening the operating bandwidth and output power compared to linear 

piezoelectric harvesters. However, under low intensity vibrational, the bi-stable piezoelectric 

harvester may struggle to overcome potential energy barriers for substantial inter-well 

motion [17-20]. Consequently, researchers proposed nonlinear multi-stable piezoelectric 

harvesters, typically utilizing multiple magnets to create multiple potential wells. By 

adjusting system parameters, the barrier height between wells can be reduced, making it 

easier for the system to generate high-energy inter-well motion. Wang et al. [21] studied 

the influence mechanism of geometric nonlinearity and gravitational effects on the 

dynamic response of a tri-stable piezoelectric harvester through theoretical analysis and 

experimental methods. Zhu et al. [22] analyzed the impact of three potential well 

distribution characteristics on the energy harvesting performance of tri-stable piezoelectric 

harvesters, showing optimal performance when the barrier heights are equal. Yang et al. 

[23] investigated a hybrid tri-stable vibration energy harvester employing geometric 

nonlinearity to adjust the resonant frequency and expand the inter-well motion bandwidth. 

The results indicate that the tri-stable characteristic has the potential to enhance significantly 

the harvested power. Zhou et al. [24] proposed penta-stable cantilever beam piezoelectric 

harvesters to achieve greater deformation and higher output voltage. 

Most studies on piezoelectric harvesters have focused on harmonic excitation. However, 

rotational motion is prevalent in both civilian and industrial environments. In recent years, 

some researchers have focused on using piezoelectric harvesters to achieve self-powering 

for sensor nodes in rotating machinery such as wheels and wind turbines [25,26]. 

Khameneifar et al. [27] designed a cantilever beam linear piezoelectric harvester that 

generates continuous vibration and harvests energy through the periodic excitation caused 

by the tip mass of the beam during rotational motion. Zhang et al. [28] proposed a bi-stable 

piezoelectric harvester mounted on the edge of a hub, capable of converting the rotational 

and kinetic energy generated by external excitation into electrical energy to power a tire 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 259 

pressure monitoring system. Mei et al. [29] established a theoretical model of a magnetic 

tri-stable piezoelectric harvester based on rotational motion, analyzing the impact of different 

speeds and piezoelectric coupling coefficients on the system's energy harvesting efficiency. 

Currently, effective broadening of the operating frequency band and improvement of 

the power performance of piezoelectric harvesters under different rotational radius and 

speeds remain urgent research issues. This paper designs a hybrid tri-stable piezoelectric 

harvester (HTPEH) by attaching a novel dynamic amplifier composed of rotational and 

vertical springs to the fixed end of a traditional cantilever piezoelectric beam. This 

amplifier simultaneously increases the rotational amplitude and vertical displacement of 

the cantilever beam during rotational motion. The influence of rotational radius, magnetic 

mass at the beam's end, and spring stiffness ratio on the steady-state dynamic response of 

the HTPEH under rotational motion is investigated. 

2. MODELING OF HYBRID TRI-STABLE PIEZOELECTRIC ENERGY HARVESTER 

Figure 1 depicts a HTPEH mounted on the edge of a vehicle wheel hub, with r being 

the radius of the wheel hub. In Fig. 2, a schematic illustration of the HTPEH during 

rotational motion is given. It features a piezoelectric cantilever beam, with l and b denoting 

the length and width of the beam, respectively. The free end of the beam is equipped with 

a tip magnetic denoted as A and a mass mf is attached to the fixed end through rotational 

spring kr and vertical spring kf connecting the wheel hub edge and a L-shaped frame. The 

mass mf combined with springs kr and kf constitutes a novel power amplifier, capable of 

simultaneously increasing the rotational amplitude and vertical displacement of the 

cantilever beam during rotational motion. The piezoelectric beam consists of a substrate 

layer with thickness hs and two piezoelectric layers covering its upper and lower surfaces. 

The thickness of the substrate layer and piezoelectric layer are hs and tp. The external load 

resistance RL is connected in series. Two external magnets, denoted in the figure by B and 

C, are arranged on the L-shaped frame, and their appropriate spacing with the tip magnetic 

tip induces a tri-stable potential energy function for the system. 

 

Fig. 1 The HTPEH installed on a vehicle wheel hub 



260 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

z(z )

y

xpiezoelectric layer

substrate layer

 

Fig. 2 The schematic diagram of the HTPEH in rotational motion 

In Fig. 2, the x-y-z coordinates represent the fixed coordinate system, and x*-y*-z* is 

the rotating coordinate system. Assuming R0 defines the position of an infinitesimal 

segment dx on the beam in the rotating coordinate system, w(x,t) and u(x,t) denote the 

lateral and axial displacements at position x during beam vibration. The angle between the 

x-axis and the x*-axis is defined by 𝜃 = ∫ 𝜔𝑑𝑡
𝑡

0
, where ω is the angular displacement of the 

wheel hub. The eccentricities of the tip magnetic and the mass block mf are denoted as et 

and ef, respectively. The constitutive relationships for the metal base layer and piezoelectric 

layers of the piezoelectric beam are as follows: 

 

s s

1 s 1

p p

1 p 1 31 3

T

3 31 1 33 3

( )

T Y S

T Y S d E

D d T ε E

=


= − 


= + 

 

 

(1) 

Here, the subscript/superscript s represents the parameters associated with the substrate 

layer, while the subscript/superscript p represents those associated with the piezoelectric 

layer. The x* and y* directions are indicated by subscripts 1 and 3, respectively. T, S, and 

Y represent the stress, strain, and Young's modulus of the beam, respectively. D3, d31, and 

휀33
𝑇  denote the electric displacement, piezoelectric constants, and dielectric constants, 

respectively. With V(t) denoting the voltage, the relation 𝐸3 = −𝑉(𝑡)/(2𝑡𝑝) holds, where 

E3 is the electric field. The relationship between the displacement and strain is expressed 

as 𝑆1
𝑠 = 𝑆1

𝑝
= −𝑦𝑤", where y represents the distance between the neutral axis of the beam 

and the point on its cross-section. 

The Lagrangian for the HTPEH is expressed as follows: 

 eWUTL +−=
 

(2)
 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 261 

where T represents the kinetic energy of the system, U is the overall potential energy 

consisting of the strain energy, Ue, the gravitational potential energy, Ug, the centrifugal 

potential energy Uc, the strain energy, Ud, and the magnetic potential energy Um. Finally, 

We represents the electrical energy of the system. Based on the geometric relationships 

presented in Fig. 1(b), the kinetic energy of the system can be expressed as: 

 

( ) 
( ) 

( ) 

2
2 2

0
0

2
2 2

f f

2
2 2

t t

2 2

f t

1
( , ) ( , ) ( , ) d

2
1

(0, ) (0, ) (0, ) (0, )
2
1

( , ) ( , ) ( , ) ( , )
2
1 1

(0, ) ( , )
2 2

l

T m w x t w x t x r u x t x

m w t w t e w t r u t

m w l t w l t e w l t l r u l t

J w t J w l t

 

 

 

 = + + + + 

 + + − + + 

 + + + + + + 

 + +



  

(3)

 

The strain energy of the system can be expressed as: 

 2

e p 31 s p
0

1 1
( , ) ( ) ( ) ( , ) d

2 2

l

U YIw x t Y bd h t V t w x t x
 

 = − + 
 

   

(4)

 

where 𝑌𝐼 =
2

3
[𝑌𝑠𝑏ℎ

3 + 𝑌𝑝𝑏(3ℎ
2𝑡𝑝 + 3ℎ𝑡𝑝

2 + 𝑡𝑝
3)] represents the bending stiffness of the 

beam.  

The gravitational potential energy of the system is given by: 

     g 0 f t
0

( )sin ( , ) cos d g sin (0, )cos g ( )sin ( , ) cos
l

U m g x r w x t x m r w t m l r w l t     = + + + + + + +
 
(5)

 

The centrifugal potential energy of the system in rotational motion is given by: 

 

2 2 2 2 2 2 2

c 0 0 f t
0

1 1
[ ( ) ( ) ( )] ( , ) d

2 2

l

U m r l x m l x m r m l r w x t x    = − + − + + +
  

(6)
 

The strain potential energy of the dynamic amplifier is given by: 

 

2 2

d r f

1 1
(0, ) (0, )

2 2
U k w t k w t= +

 
 (7)

 

The electrical energy of the system is given by: 

 𝑊𝑒 =
1

4
𝑌𝑝𝑏𝑑31(ℎ𝑠 + 𝑡𝑝)𝑉(𝑡) ∫ 𝑤″(𝑥, 𝑡)𝑑𝑥

𝑙

0
+ 𝑏𝑙휀33

𝑠 𝑉(𝑡)2

4𝑡𝑝
  (8) 

where, 휀33
𝑠 = 휀33

𝑇 − 𝑑31
2 𝑌𝑝.  

Utilizing the Galerkin method, the latera displacement w(x,t) is expressed as: 

 1

( , ) ( ) ( )
n

r r

r

w x t x q t
=

=
  

 (9)

 

Here, 𝜙𝑟(𝑥)  and qr(t) represent the rth order modal shape function and generalized 

coordinate of the beam, respectively. The orthogonality condition for the modal shape 

functions is: 

2

0 t t t t t t t t
0

2

f f f f f f f f

( ) ( )d ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

(0) (0) (0) (0) (0)( ) (0) (0) (0)

l

s r s r s r s r s r

s r s r s r s r rs

s m s s l m l l m e l l J m e l l m e l

m m e J m e m e

         

        

   + + + + +

   + − + + − =
  (10) 



262 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

 

2 2

2

R f2 20

d ( ) d ( )
d (0) (0) (0) (0)

d d

l
s r

m r m r r rs

s s
YI s k k

s s

 
      + + =

  (11) 

where, 𝛿𝑟𝑠 is the Kronecker delta function. The natural frequency of undamped vibration 

for the beam is given by 𝜔𝑟 = 𝜆𝑟
2√𝑌𝐼 (𝑚𝑙4⁄ ) , where 𝜆𝑟  is the eigenvalue. The 

computational method for 𝜆𝑟 and the mode shape functions has been presented by Man et 

al. [30]. Considering only the first-order mode, based on the author's previous research, the 

expression for magnetic potential energy, accounting for the eccentricity of the magnet at 

the beam tip, is given by: 

 

2 4 6 7

m 0 1 1 2 1 3 1 1

1 1 1
( )

2 4 6
U k k q k q k q q= − + + +

  

(12) 

The expressions for the coefficients in the equation can be found in reference [30]. 

Considering only the first-order mode, substituting Eq. (9) into Eq. (2) and then into the 

following Lagrangian variational equation: 

 

d
( )

dt

d
( )

dt

L L
P t

q q

L L
Q t

VV

   
− =  

   


  
− =        

(13)

 

where, 𝑃(𝑡) = −2𝜉1𝜔1�̇�1(𝑡) represents the generalized dissipative force of the system, ω1 

is the first-order natural frequency of the system, 𝜉1 is the damping ratio of the system, Q(t) 

represents the generalized output charge, and �̇�(𝑡) = −𝑉(𝑡)/𝑅𝐿, Utilizing Eq. (13), the 

electromechanical equation of the HTPEH can be derived as follows: 

 2 2 3 5

1 1 1 1 1 1 c 1 1 1 2 1 3 12 g ( ) cosq q q K q k q k q k q V F t      + + + − + + + − = − +
 

 
(14)

 

 
p

L

0
V

C V q
R

+ + =

 
 (15)

 

In Eqs. (14) and (15), the expressions for the coefficients Kc, 𝜒, 𝜗, 𝛤 and Cp can be found 

in the appendix, while F(t) represents the external excitation arising from the interaction 

between the vehicle wheel and the road surface. 

By introducing dimensionless parameters 𝑆(𝜏) = 𝑞1(𝑡)/𝑙, 𝜏 = 𝜔1𝑡, �̄�(𝜏) = 𝑉𝐶𝑝/(𝑙𝜗) 

into Eqs. (14) and (15), we obtain: 

 

2 3 5

1 1 c 2 32 (1 ) ( ) cosS S K K S K S K S V g f t    + + − + + + + − =  +    
(16)

 

 0V V S+ + =    (17) 

where, 𝐾1 = 𝑘1/𝜔1
2 , 𝜔 = �̇�/𝜔1 , 𝐾2 = 𝑘2𝑙

2/𝜔1
2，𝐾3 = 𝑘3𝑙

4/𝜔1
2 , �̄� = 𝜒/𝑙 , �̄� = 𝛤/

(𝑙𝜔1
2), 𝑓(𝑡) = 𝐹(𝑡)/(𝑙𝜔1

2),𝛩 = 𝜗2/(𝐶𝑝𝜔1
2). 

This study primarily investigates the rotational motion performance of HTPEH under 

various constant speeds in the absence of external excitations. Therefore, let �̈� = 0, 𝑓(𝑡) = 0, 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 263 

considering the system's own gravity �̄�𝑔 as an external excitation force F, then Eq. (16) 

can be rewritten as: 

 
2 3 5

1 1 c 2 32 (1 ) cosS S K K S K S K S V F  + + − + + + − =
  

(18) 

3. THEORETICAL ANALYSIS BY HARMONIC BALANCE METHOD 

The solutions of Eqs. (18) and (17) may be assumed in the following form: 

 

( ) ( )sin( ) ( )cos( )

( ) ( )sin( ) ( )cos( )

S A B

V C D

    

    

= +


= +  

 (19)

 

Substituting Eq. (19) into Eq. (17) and balancing the coefficients of the sin(ωτ) and 

cos(ωτ) terms, for steady-state motion, setting all time derivatives to zero, yields: 

 

( )

( )

2 2

2 2

C B A

D A B


 

 


 

 


= − +


 = − −
 +   

(20)

 

Substituting Eqs. (19) and (20) into Eq. (18), and repeating the above calculations while 

neglecting higher-order harmonic terms, we obtain: 
2

2 2 3 2 5 3 2 432

1 c 2 2 2 2

53
2 (1 ) ( ) ( 2 ) 0

4 8

KK
A B K K A A AB A A B AB B A

 
   

   

 
− − + − + + + + + + − + =

+ +  
(21)

 

2
2 2 2 3 4 2 3 532

1 c 2 2 2 2

53
2 (1 ) ( ) ( 2 )

4 8

KK
B A K K B A B B A B A B B A B F

 
   

   

 
− + + − + + + + + + + + =

+ +  
(22)

 

Let the displacement amplitude of the system's steady-state motion be 𝑎 = √𝐴2 + 𝐵2, 

The voltage amplitude is 𝑢 = √𝐶2 + 𝐷2. Squaring both sides of Eqs. (21) and (22), and 

adding them together, we obtain: 

 

( )
2 22

2 2 2 2 4 2 232

1 c 2 2 2 2

53
1 2

4 8

KK
a K K a a a F

 
   

   

  
− + − + + + + + + =  + +   

 (23) 

Therefore, the steady-state displacement amplitude of the HTPEH in rotational motion 

can be determined by Eq. (23), and the expression for the steady-state output voltage 

amplitude is 

 
2 2

u a


 
=

+   

(24) 

4. DYNAMIC RESPONSE ANALYSIS 

In the first part of this section, analytical solutions for the HTPEH system are obtained 

using the harmonic balance method. The study investigates the influence of parameters 

such as the masses of the two ends of the beam, rotational radius, rotational speed, and the 



264 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

stiffness ratio of the springs on the steady-state response of the HTPEH system during 

rotational motion. The correctness of the analytical solutions is verified through frequency 

sweep simulations. 

In the second part, the system's energy harvesting performance in the time domain is 

analyzed using the ode45 solver in MATLAB. The geometric and material parameters for 

the HTPEH are given in Table 1. 

Figures 3-5 depict the variations in steady-state displacement and output voltage 

amplitude of the HTPEH compared to the TPEH with different values of the mt as the 

rotational speed changes when mt = 25 g, mf = 100 g, r = 0.21 m. The solid lines represent 

the analytical solutions of the HTPEH, the scattered points represent the analytical 

solutions of the TPEH, and the shaded regions represent the numerical solutions obtained 

by rotational speed sweeping of the HTPEH. From Fig. 3, it can be observed that both 

HTPEH and TPEH can generate significant inter-well motion during rotational motion 

influenced by their respective gravity. Compared to TPEH, HTPEH exhibits a significantly 

higher peak value of output voltage and inter-well motion range. As seen from Figs. 3 to 

5, with the increase of the mt, the displacement, output voltage and inter-well motion range 

of both piezoelectric harvesters increase significantly, and the energy harvesting advantage 

of HTPEH becomes more significant. When mt = 25 g, the peak voltage and inter-well motion 

range of HTPEH increase by 11.85% and 29.26%, respectively, compared to TPEH. When 

mt = 27.5 g, these two indicators increase to 11.95% and 29.50%, respectively, and when mt 

= 30 g, they further increase to 12.34% and 30.42%. The larger mt, the more significant the 

increase in peak voltage and inter-well motion range. Moreover, it can be observed that the 

numerical simulation results of the steady-state displacement and output voltage amplitude 

of HTPEH under different rotational speeds by rotational speed sweep are in good agreement 

with the analytical solutions obtained by the harmonic balance method. 

Table 1 The geometry and material parameters of HTPEH 

Parameters Symbol Value 

Piezoelectric film length  l 80 mm 

Piezoelectric film width  b 20 mm 

Piezoelectric film thickness  hs 0.2 mm 

Young's modulus of the metal substrate Ys 70 GPa 

Density of the metal substrate s 2700 kg/m3 

Density of the piezoelectric layer p  7750 kg/m3 

Magnetization intensity  MA, MB, MC 1.22×106 A/m 

Magnetic volume VA, VB, VC 1.0×10-6 m-3 

Damping ratio 1 0.01 

Young's modulus of the piezoelectric layer  Yp 60.98 GPa 

Piezoelectric strain constant  d31 -1.71×10-10 C/N 

Dielectric constant of the metal substrate s
33 -1.33×10-8 F/m 

Transverse distance of the magnet dh 18 mm 

Vertical distance of the magnet  dv 9 mm 

 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 265 

 

Fig. 3 a) Dependency of the dimensionless displacement and b) voltage amplitude of the 

system on the rotational speed for different models and mt = 25 g 

  

Fig. 4 a) Dependency of the dimensionless displacement and b) voltage amplitude of the 

system on the rotational speed for different models and mt = 27.5 g 

  

Fig. 5 a) Dependency of the dimensionless displacement and b) voltage amplitude of the 

system on the rotational speed for different models and mt = 30 g 



266 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

Figure 6 presents the curves of steady-state displacement and voltage amplitude with 

rotational speed for different values of the spring stiffness ratio �̄� = 𝑘𝑓/𝑘𝑟, when mt = 25 g 

and r = 0.21 m. It can be observed from Fig. 6 that the output voltage-rotational speed 

response curves for the three kf/kr ratios are essentially overlapping, while the steady-state 

output displacement, peak voltage and rotational speed range of inter-well motion all 

increase as kf/kr decreases. 

  

Fig. 6 a) Dependency of the dimensionless displacement and b) voltage amplitude response 

curves on the rotational speed for different spring stiffness ratios 

To investigate the time-domain dynamic performance of HTPEH and TPEH at different 

wheel hub rotational speeds, Fig. 7 shows the phase diagram and voltage time-history 

diagram for both piezoelectric harvesters at 300 rpm when mt = 25 g, mf = 100 g, r = 0.21 m. 

It can be observed from Fig. 7 that HTPEH quickly transitions from intra-well motion to 

significant inter-well motion and generates a high output voltage after a short period of 

intra-well motion. In contrast, TPEH cannot overcome the potential barrier, only exhibiting 

small oscillations within the intra-wel, resulting in a small output voltage. This indicates 

that at the same rotational speed, HTPEH is more likely to escape the potential well and 

enter significant inter-well motion. 

As the rotational speed increases to 350 rpm, it can be seen from Fig. 8 that HTPEH 

instantly enters a high-energy orbit to perform substantial inter-well motion, generating a 

large output voltage. On the other hand, TPEH, after a period of chaotic motion, eventually 

overcomes the potential barrier and escapes the well, entering significant inter-well motion 

with a longer time duration and a lower output voltage than HTPEH. In summary, at lower 

rotational speeds, HTPEH is more likely to break free from the potential well, generating 

higher output voltages and capturing more energy during rotational motion, demonstrating 

the advantages of HTPEH in energy harvesting performance. 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 267 

 

  

Fig. 7 (a) Three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage 

time-history diagram for HTPEH and TPEH with the rotational speed of 300 rpm 

Figure 9 presents the curves of the steady-state displacement amplitude and steady-state 

output voltage amplitude of HTPEH at different rotational radii with varying speeds when 

mt = 25 g, mf = 100 g. It can be observed that when the rotational speed is below 100 rpm, 

the change in rotational radius has a minimal impact on the steady-state displacement 

output voltage amplitude of HTPEH. However, when the speed exceeds 100 rpm, both the 

steady-state displacement output voltage amplitude decrease with an increase in the 

rotational radius. Moreover, as the rotational radius becomes larger, the decrease in 

amplitude becomes more pronounced.  

Taking the peak value of the inter-well motion displacement under different rotational radii 

as an example, when the rotational radius is between 0.18 m and 0.33 m, the peak voltage of 

HTPEH shows a slow decline with an increase in rotational radius, and the corresponding inter-

well motion range slowly increases. When the rotational radius is between 0.33 m and 0.37 m, 

the peak voltage of HTPEH sharply decreases with an increase in the rotational radius, and the 

corresponding inter-well motion range sharply increases. When the rotational radius is greater 

than 0.33 m, both the displacement and voltage peak values are observed to be at a very low 

level. This is because at this rotational radius, the HTPEH is restricted to intra-well motion. 



268 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

 

 

Fig. 8 (a) Three-dimensional phase diagram, (b) two-dimensional phase diagram, and (c) voltage 

time-history diagram for HTPEH and TPEH with the rotational speed of 350 rpm 

 

Fig. 9 Dependency of the dimensionless a) displacement amplitude and (b) the voltage 

amplitude on rotational speed for HTPEH and different rotational radii 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 269 

Although increasing the rotational radius leads to a decrease in the peak voltage of 

HTPEH, it simultaneously increases the range of inter-well motion. Consequently, the 

system can capture energy from rotational motion over a broader speed range. Therefore, 

an appropriate radius should be set based on practical considerations such as the installation 

position, the dimensions and the specific application requirements of HTPEH. 

Figure 10 depicts the phase portraits and voltage-time history diagram for HTPEH at a 

rotational speed of 300 rpm, with wheel hub radii corresponding to r = 0.18 m and 

r = 0.21 m. It can be observed that when r = 0.18 m, the system undergoes a period of 

chaotic motion, subsequently overcoming the potential barrier to perform significant inter-

well motion and generating a higher output voltage. At r = 0.21 m, the system cannot 

overcome the potential barrier, remaining in small oscillations within one well and unable 

to enter inter-well motion, resulting in only a small output voltage. As the rotational 

increases to 350 rpm, as shown in Fig. 11, HTPEH under both r = 0.18 m and r = 0.21 m 

can generate inter-well motion. However, the system with the smaller rotational radius 

requires less time to enter inter-well motion. When the system with a rotational radius of  

 

  

Fig. 10 (a) Three-dimensional phase diagram, (b) two-dimensional phase diagram, and 

(c) voltage-time history diagram for HTPEH with different values of wheel hub 

radius at a rotational speed of 300 rpm 



270 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

r = 0.21 m enters stable inter-well motion, its output voltage is significantly lower than that 

of the system with the rotational radius of r = 0.18 m. Figures 9 and 10 illustrate that 

reducing the rotational radius or increasing the rotational speed can make the HTPEH more 

easily enter substantial inter-well motion, resulting in stable output voltage. 

 

  

Fig. 11 (a) Three-dimensional phase diagram, (b) two-dimensional phase diagram, and 

(c) voltage-time history diagram for HTPEH with different values of wheel hub 

radius at a rotational speed of 350 rpm 

Figures 12 and 13 show the output power amplitude curves of the system at various 

rotational speeds with different load resistances for rotational radii of r = 0.21 m and 

r = 0.36 m, respectively. It is evident that for HTPEH with different rotational radii, 

substantial inter-well motion can only be generated within a certain rotational speed range. 

Beyond a certain rotational speed, the system transitions into intra-well motion, resulting 

in a very small output power, and the rotational speed range conducive to inter-well motion 

generation expands with an increase in the wheel hub radius. 



 Nonlinear Dynamical Characteristics of Hybrid Tri-Stable Piezoelectric Energy Harvester Based... 271 

 

Fig. 12 a) The variation surface and b) top view of the impact of resistance on the system's 

output power at the wheel hub radius of 0.21 m 

 

Fig. 13 a) The variation surface and b) top view of the impact of resistance on the system's 

output power at the wheel hub radius of 0.36 m 

5. CONCLUSION 

This paper introduces a hybrid tri-stable piezoelectric energy harvester based on 
rotational motion. The dynamic model of the system is established using Lagrangian 
equations. The influence of various factors such as the mass of the piezoelectric beam tip 
magnet, wheel hub radius, rotational speed, and the stiffness ratio of the dynamic amplifier 
on the system's steady-state dynamic response is analyzed. A comparison with traditional 
tri-stable piezoelectric energy harvesters is conducted. The theoretical analysis is validated 
through rotational sweep numerical simulations, and the system's time-domain dynamic 
performance is analyzed using the MATLAB software. The main conclusions from the 
theoretical analysis and numerical simulations are as follows: 

▪ Compared to traditional tri-stable piezoelectric energy harvesters without a dynamic 
amplifier, the hybrid tri-stable piezoelectric energy harvester exhibits significantly 
improved output voltage and inter-well motion range during rotational motion. 

▪ At lower speeds, changes in wheel hub radius minimally affect the HTPEH system's 
output voltage. While the range of inter-well motion expands with an increase in 
wheel hub radius, but there is a concurrent decrease in peak output voltage. 



272 D.-W. MAN, Y. ZHANG, L.-P. TANG, Q.-H. XU, D. CHEN, ET AL. 

▪ Increasing the mass of the tip magnet or reducing the stiffness ratio of the vertical 

spring to the rotational spring can simultaneously increase the HTPEH system's 

output voltage amplitude and range of inter-well motion. 

▪ At higher rotational speeds or smaller wheel hub radii, the HTPEH is more likely to 

enter large inter-well motion, resulting in higher output voltage. 

Acknowledgment: This research was funded by the Anhui Provincial University Provincial Natural 

Science Research Project-Key Project (2022AH050240), the Doctoral Startup Foundation of Anhui 

Jianzhu University (2020QDZ07), the Scientific research Project of Anhui Education Department-

Key Project(KJ2021JD01), the Anhui Provincial Natural Science Foundation (2108085MA28), the 

Mechanical Properties Research of FRP Connectors in Fabricated Concrete Sandwich Insulation 

Wall Panels after Fire (KJ2021A0608), the Anhui Provincial Course Ideological and Political 

Demonstration Course(2022kcsz060). 

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Appendix 

2 2 2 2 2

c 0 1 1 1
0 0 0

2 2 2 2

f 1 1 t 1 1
0 0

1
( ) ( ) d ( ) ( ) d ( ) d

2

( ) d (0) ( ) ( ) d ( )

l l l

l l

K m r l x x x l x x x x x

m r x x m l r x x l

  

   

 
 = − + − − 

 
    + − + + −
      

  

 
 

0 1 f 1 t 1 f f 1 t t 1
0
( ) ( )d (0) ( ) ( ) (0) ( ) ( )

l

m x r x x m r m l r l m e m e l r l      = + + + + − + +  

p 2

p 31 1
0

( ) d
2

lt
Y bd h x x 

 
= + 

 
 , 

0 1 t 1 f 1
0

( )d ( ) (0)
l

m x x m l m   = + + , 

S

33

p

p2

bl
C

t


=