A Practical Method for the Dynamic Analysis of Non-Uniform Piezoelectric Rod

Abstract

In this paper, a unified approach for the dynamic analysis of non-uniform piezoelectric rod is presented. It is assumed that the cross sectional area of the rod is varying along the longitudinal axis, arbitrarily. Therefore, the partial differential equations that govern the non-uniform piezoelectric isotropic rod in a forced vibration analysis are obtained with a variable coefficient taking into account mechanical and electrostatic equations. Analytical solutions of these equations are only possible for simple crosssection areas. First, the governing equations are transformed to the Laplace space and then solved numerically by pseudospectral Chebyshev approach for arbitrary cross-section area under four different load functions. The final results are transformed to the time domain using modified Durbin’s procedure. The technique is validated for simple cross-section area results that can also be solved analytically.

Keywords

• Forced vibration,Piezoelectric rod,Laplace transform,Pseudospectral Chebyshev method

Düzgün Olmayan Piezoelektrik Çubuğun Dinamik Analizi için Pratik Bir Yöntem

Abstract

Bu çalışmada, düzgün olmayan piezoelektrik çubuğun dinamik analizi için birleşik bir yaklaşım sunulmaktadır.  Çubuğun enine kesit alanının rastgele olarak uzunlamasına eksen boyunca değiştiği varsayılmaktadır.  Bu nedenle, zorlanmış titreşim analizinde düzgün olmayan piezoelektrik izotropik çubuğu idare eden kısmi diferansiyel denklemler, mekanik ve elektrostatik denklemler dikkate alınarak değişken bir katsayılı olarak elde edilirler.  Bu denklemlerin analitik çözümleri sadece basit kesit alanları için mümkündür. İlk olarak, sistemi idare eden denklemler Laplace uzayına dönüştürülür ve daha sonra dört farklı yük fonksiyonu altında rastgele kesit alanı için pseudospektral Chebyshev yaklaşımı ile sayısal olarak çözülür. Nihai sonuçlar, modifiye edilmiş Durbin prosedürü kullanılarak zaman uzayına dönüştürülür. Yöntem, analitik olarak da çözülebilen basit kesit alanına sahip piezoelektrik çubuk sonuçları ile doğrulanmıştır.

Keywords

• Zorlanmış titreşim,Piezoelektrik çubuk,Laplace dönüşümü,Pseudospectral Chebyshev yöntemi

References

1. [1] Eisenberger, M., 1991. Exact Longitudinal Vibration Frequencies of a Variable Cross-Section Rod, Applied Acoustics, 34(2) (1991), 123-130.
2. [2] Abrate, S., 1995. Vibration of Non-Uniform Rods and Beams, Journal of Sound and Vibration, 185(4) (1995), 703-716.
3. [3] Kumar, B. M., 1997. Sujith, R. I., Exact Solutions for the Longitudinal Vibration of Non-Uniform Rods, Journal of Sound and Vibration, 207(5) (1997), 721-729.
4. [4] Li, Q. S., 2000. Free Longitudinal Vibration Analysis of Multi-Step Non-Uniform Bars Based on Piecewise Analytical Solutions, Engineering Structures, 22(9) (2000), 1205-1215.
5. [5] Li, Q. S., 2000. Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section, Journal of Vibration and Acoustics, 122(2) (2000), 183-187.
6. [6] Yardimoglu, B., Levent, A., 2011. Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections, Shock and Vibration 18(4) (2011), 555-562.
7. [8] Zhang, C. L., Chen, W. Q., Li, J. Y. and Yang, J.Y., One-Dimensional Equations for Piezoelectromagnetic Beams and Magnetoelectric Effects in Fibers, Smart Materials and Structures, 18(9) (2009). [7] Chen, W. Q., Zhang, C. L., 2009. Exact Analysis of Longitudinal Vibration of a Non-Uniform Piezoelectric Rod, Second International Conference on Smart Materials and Nanotechnology in Engineering, International Society for Optics and Photonics.
8. [9] Nadal, C., François, P., 2009. Multimodal Electromechanical Model of Piezoelectric Transformers by Hamilton’s Principle, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 56(11) (2009), 2530-2543.

9. [10] Li, Y., Zhifei, S., 2009. Free Vibration of a Functionally Graded Piezoelectric Beam via State-Space Based Differential Quadrature, Composite Structures, 87(3) (2009), 257-264.
10. [11] Yıldırım, K., Küçük, I., 2016. Active Piezoelectric Vibration Control of a Timoshenko Beam, Journal of Franklin Institute, 353(1) (2016), 95-107.
11. [12] Küçük, I., Yıldırım, K., Sadek, ˙I., Adali, S., 2015. Optimal Control of a Beam with Kelvin-Voigt Damping Subject to Forced Vibrations Using a Piezoelectric Patch Actuator, Journal of Vibration And Control, 21(4) (2015), 701-713.
12. [13] Yıldırım, K., 2016. Optimal Forced Vibrations Control of a Smart Plate, Applied Mathematical Modelling, 40 (2016), 6424-6436.
13. [14] Eker, M., Celebi, K., Yarımpabuc, D., 2015. Free Vibration Analysis of Nonuniform Piezoelectric Rod by Complementary Functions Method, European Conference on Numerical Mathematics and Advanced Application, Metu, Ankara, 14-18 September.
14. [15] Yarimpabuc, D., Eker, M., Celebi, K., 2016. Free Vibration Analysis of Non-Uniform Piezoelectric Rod by Chebyshev Pseudospectral Method, 1st International Mediterranean Science and Engineering Congress, 26-28 October, Adana, 1563-1568.
15. [16] Yarimpabuc, D., Eker, M., Celebi, K., 2018. Forced Vibration Analysis of Non-Uniform Piezoelectric Rod by Complementary Functions Method, Karaelmas Science and Engineering Journal, 8(2) (2018), 496-504.
16. [17] Trefethen, L.N., 2000. Spectral Methods in Matlab, SIAM, Philadelphia, PA.
17. [18] Fornberg, B., 1996. A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge.
18. [19] Bazan, F.S.V., 2008. Chebyshev Pseudospectral Method for Computing Numerical Solution of Convection-Diffusion Equation, Applied Mathematics and Computation, 200 (2008), 537-546.
19. [20] Durbin, F., 1974. Numerical Inversion of Laplace Transforms: An Efficient Improvement to Dubner and Abrate’s Method, Computer Journal, 17 (1974), 371-376.
20. [21] Temel, B., Yıldırım, S., Tütüncü, N. 2014. Elastic and viscoelastic Response of heterogeneous annular structures under Arbitrary Transient Pressure, International Journal of Mechanical Sciences, 89 (2014), 78-83.
21. [22] Gottlieb, D., 1981. The Stability of Pseudospectral- Chebyshev Methods, Mathematics of Computation, 36(153) (1981), 107-118.
22. [23] Ding, H. J., Chen, W. Q. 2001. Three Dimensional Problems of Piezoelasticity, Nova Science Publishers, New York, USA.