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Analysis of nonlinear vibration response of a functionally graded truncated conical shell with piezoelectric layers

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Abstract

The nonlinear vibration response of a functionally graded materials (FGMs) truncated conical shell with piezoelectric layers is analyzed. The vibration amplitude is suppressed by the positive and inverse piezoelectric effects. And the bifurcation phenomenon is described to reveal the motion state of the conical shell. Firstly, a truncated conical shell composed of three layers is described. And the effective material properties of the FG layer are defined by the Voigt model and the power law distribution. Next, the electric potentials of piezoelectric layers are defined as cosine distribution along the thickness direction. Meanwhile, the constant gain negative velocity feedback algorithm is used to suppress the vibration amplitude by the electric potential produced by the sensor layer. Thereafter, considering the first-order shear deformation theory and the von Karman nonlinearity, the relationship between the strain and displacement is defined. And the corresponding energy of the conical shell is calculated. After that, the motion equations of the conical shell are derived based on the Hamilton principle. Again, the nonlinear single degree of freedom equation is derived by the Galerkin method and the static condensation method. In the end, the nonlinear vibration response of FGMs truncated conical shell with piezoelectric layers under the external excitation is analyzed via using the harmonic balance method and the Runge-Kutta method. The effects of various parameters, such as ceramic volume fraction exponent, external excitation’s amplitude, control gain and geometric parameters on the nonlinear vibration response of the system are evaluated by case studies. Results indicate that the control gain plays an important role on the suppression of the vibration amplitude. The ceramic volume fraction exponents are not sensitive to the nonlinear vibration response compared with other parameters. The bifurcation behavior is observed under different parameters. The FGMs truncated conical shell with piezoelectric layers has three types of motion state, such as periodic motion, multi-periodic motion, and chaos motion.

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Abbreviations

x :

Axial coordinate

θ :

Circumferential coordinate

z :

Radial coordinate

α 0 :

Semi-vertex angle

L :

Length of the conical shell

R 1 :

Radius of small end of the shell

R 2 :

Radius of large end of the shell

h s :

Thickness of sensor layer

h f :

Thickness of FG layer

h a :

Thickness of actuator layer

Γ :

Material property

Γ 1, Γ 2, Γ 3 :

Temperature sensitivity coefficients

T :

Temperature

Γ m, Γ c :

Material property of metal and ceramic

V m, V c :

Volume fractions of metal and ceramic

E xa, E ap, E za :

Components of electric field of actuator layer

E xs, E as, E zs :

Components of electric field of sensor layer

ϕ a, ϕ s :

Electric potential functions of actuator and sensor layers

φ a, φ s :

Electric potentials at the middle surface of actuator and sensor layers

z a, z s :

Local thickness coordinate of actuator and sensor layers

V :

External applied control voltage

G :

Control gain

H x, H θ, H z :

Lame coefficients

u, v, w :

Displacement components of any point

u 0, v 0, w 0 :

Displacement components on the middle surface

Φ x, Φ θ :

Transverse normal rotations

t :

Time variable

ε x, ε θ, γ xθ, γ xz, γ θz :

Strain components

σ x, σ θ, τ x θ, τ θz, τ xz :

Stress components

Q i j :

Equivalent stiffness

U s, U a :

Electric enthalpy

U f :

Strain energy

K :

Kinetic energy

W :

Work done by external load

F :

External excitation

Ω :

Excitation frequency

N :

Internal force

M :

Bending moment

Q :

Shear force

I :

Inertia coefficient

m :

Axial wave numbers

n :

Circumferential wave numbers

f :

Excitation amplitude

\(\bar w\), τ :

Dimensionless coefficients

χ :

Dimensionless amplitude

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Acknowledgments

This research work is supported by the National Natural Science Foundation of China (No. 51965042), the Postgraduate Innovation Fund Project of Nanchang Hangkong University (No. YC2021016).

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Authors and Affiliations

  1. School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang, Jiangxi Province, 330063, China

    Yuhang Zhang, Wenguang Liu, Zhipeng Lyu & Lei Pang

  2. Graduate School at Shenzhen, Harbin Institute of Technology, Harbin, 150001, China

    Chao Liu

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  1. Yuhang Zhang
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  2. Wenguang Liu
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Correspondence to Wenguang Liu.

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Wenguang Liu received his Ph.D. degree from Nanjing University of Aeronautics and Astronautics, China. He is currently working as Full Professor in Nanchang Hangkong University, China.

Yuhang Zhang is currently studying as a graduate student in Nanchang Hangkong University, China. His research interests include the nonlinear vibration analysis of functionally graded smart structures.

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Zhang, Y., Liu, W., Lyu, Z. et al. Analysis of nonlinear vibration response of a functionally graded truncated conical shell with piezoelectric layers. J Mech Sci Technol 36, 3897–3909 (2022). https://doi.org/10.1007/s12206-022-0712-7

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  • DOI: https://doi.org/10.1007/s12206-022-0712-7

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