Parametric Analysis of a Rotational Piezoelectric-Coupled Tapered-Bimorph Structure with Various Boundary Conditions Under Transient Axial Loading

Abstract

Purpose

The steady-state parametric analysis of a piezoelectric-coupled rotating tapered-bimorph system consists of a central host beam and two piezoelectric ceramic patches above and below it, tapering along positive X-direction and rotating in the X–Y plane, subjected to axial dynamic as well as static loading at its one end is done. The influence of three boundary conditions and various system parameters are also studied.

Methods

The mathematical modeling of the rotating tapered-bimorph system is formulated with the help of Hamilton's equation. The parametric analysis of the axially loaded system is done for clamped-free (C-F), pinned-pinned (P-P), and clamped-pinned (C-P) boundary conditions. A number of parametric instability regions along with static buckling load plots are obtained and analyzed for various values of the piezoelectric patches to the host beam thickness ratio, taper parameter, and rotational frequency using MATLAB software and depicted through a series of diagrams.

Results

The results illustrate that the principal modal frequencies of the tapered-bimorph rotating system are increased rapidly with an increase in the thickness ratio, whereas increase less significantly with a rise in taper parameter and spinning speed for both the static and dynamic loading conditions.

Conclusion

The lowest first modal frequency is observed for the P-P system under any given system configurations and operating conditions, which can be used for low-frequency rotational vibration energy harvester over the orthodoxly used C-F system. Designing low-frequency rotational vibration energy harvester and rotor blades can be done by selecting appropriate dimensions and parameters presented in this article.

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Abbreviations

\(A_{p} \left( x \right)\) :

Cross-sectional area of the PZT patch

\(A_{h} \left( x \right)\) :

Cross-sectional area of the substrate beam

b :

Widths of the structure at the hub side

\(b\left( x \right)\) :

Widths of the structure at \(x^{\prime} = B_{0} + x\)

\(B_{0}\) :

Radius of the revolving hub

\(E_{h}\) :

Young's modulus of the substrate material

\(E_{p}\) :

Young's modulus of the PZT patch

\(I_{h} \left( x \right)\) :

Area moment of inertia of the substrate beam at standard section

\(I_{p} \left( x \right)\) :

Area moment of inertia of the PZT patch at standard section

l :

Total span of the beam

N :

Spinning speed of the system

P :

Maximum amplitude of the transient load

\(P_{s}\) :

Static load

\(t_{p}\) :

Thickness of the PZT patches

\(t_{h}\) :

Thickness of the host beam

T :

Total kinetic energy of the system

U :

Total potential energy of the system

\(W_{e}\) :

Work-done due to external axial load

\(W_{c}\) :

Work-done due to centrifugal force

\(w\left( {x,t} \right)\) :

The transverse displacement

\(\eta\) :

\(= {x \mathord{\left/ {\vphantom {x l}} \right. \kern-\nulldelimiterspace} l}\)

\(\zeta\) :

\(= {w \mathord{\left/ {\vphantom {w l}} \right. \kern-\nulldelimiterspace} l}\)

\(\alpha\) :

Taper parameter

\(\rho_{h}\) :

Mass density of the host beam

\(\rho_{p}\) :

Mass density of the PZT patches

\(\omega\) :

Driving frequency

\(\omega_{n}\) :

System natural frequency

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Chand, R.R., Tyagi, A. Parametric Analysis of a Rotational Piezoelectric-Coupled Tapered-Bimorph Structure with Various Boundary Conditions Under Transient Axial Loading. J. Vib. Eng. Technol. (2021). https://doi.org/10.1007/s42417-020-00272-9

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Keywords

  • Parametric rotational vibration
  • Tapered-bimorph structure
  • PZT-coupled beam
  • Static and dynamic loading