# An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

## Keywords:

Rotor Dynamics;, differential quadrature finite elements method;, differential quadrature hierarchical finite elements method;, hp-version of FEM;, on-board shaft

## Abstract

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

## Author Biographies

### Saimi Ahmed, Mechanical Systems & Materials Engineering Laboratory, University Abou Bekr Belkaid Tlemcen, Algeria

Saimi Ahmed obtained his Ph.D in Mechanics of Materials and Structures from the University of Tlemcen, Algeria, in 2017. He is currently a Senior Lecturer at the National High School of Hydraulics Blida, Algeria. A researcher member of Mechanical Systems and Structural Engineering Laboratory, IS2M/UABT. His research interests are: Finite element methods, Structural vibration, Structural dynamics, Dynamics of rotors, Dynamics of rotating machines, computational mechanics, FG materials, Composite materials.

### Hadjoui Abdelhamid, Mechanical Systems & Materials Engineering Laboratory, University Abou Bekr Belkaid Tlemcen, Algeria

Hadjoui Abdelhamid obtained his Ph.D in mechanical engineering from the University of Tlemcen, Algeria. He is currently a professor at the University of Tlemcen, Algeria (UABT). Research Director in Mechanical Systems and Structural Engineering Laboratory, IS2M/UABT. His research interests are: Materials Engineering, Structural Engineering, Mechanical Engineering, Structural Analysis, Finite elements Modeling, Structural Dynamics, Simulation, Dynamic Analysis, Modal Analysis, Structural Vibration, Vibration Analysis.

### Bensaid Ismail, Mechanical Systems & Materials Engineering Laboratory, University Abou Bekr Belkaid Tlemcen, Algeria

Bensaid Ismail received his B.Sc, M.Sc and Ph.D degrees in Mechanical Engineering from Abou Beckr Belkaid University Tlemcen, Algeria. He is currently working in the level of the Mechanical engineering department at the same University. Dr. Bensaid does research in Mechanical and structural Engineering, Materials, Composite, Maintenance, Nanostructures and Dynamical Systems. He, as an author/co-author, has published more than 18 articles in various journals.

### Fellah Ahmed, Mechanical Systems & Materials Engineering Laboratory, University Abou Bekr Belkaid Tlemcen, Algeria

Fellah Ahmed obtained a Ph. D in rehabilitation and reliability of structures and mechanical equipment from the University of Tlemcen, Algeria, in 2019. He is currently a research associate in the laboratory of engineering of mechanical systems and structures, IS2M/UABT. His research interests are: Finite element methods, Structural vibration, Structural dynamics, Dynamics of rotors, Dynamics of rotating machines, computational mechanics, FG materials, Composite materials.

## References

Babuska and B.Q, G., 1992. The h, p and h-p version of the finite element method: basis theory and applications. Advances in Engineering Software, 15(3–4), pp. 159–174.

Babuska, I., B A, S. and I N, K., 1981. The p-Version of the Finite Element Method. SIAM Journal on Numerical Analysis, 18(3), pp. 515–545.

Babuška, I. and Suri, M., 1990. The p- and h-p versions of the finite element method, an overview. Computer Methods in Applied Mechanics and Engineering, 80(1–3), pp. 5–26.

Bardell, N. S., 1996. An engineering application of the hp version of the finite element method to the static analysis of a Euler-Bernoulli beam. Computers & structures, 59(2), pp. 195–211.

Belhadj, A., Boukhalfa, A. and Belalia, S. A., 2020. Free vibration investigation of single walled carbon nanotubes with rotary inertia. Nanomaterials Science & Engineering, 2(3), pp. 103–112.

Bellman, R. and Casti, J., 1971. Differential quadrature and long term integration. Journal of Mathematical Analysis and Applications, Volume 34, pp. 235–238.

Bert, C. and M, M., 1996. The differential quadrature method for irregular domains and application to plate vibration. International Journal of Mechanical Sciences, 38(6), pp. 589–606.

Bert, C. W. J. S. K. a. S. A. G., 1988. Two new approximate methods for analyzing free vibration of structural components. AIAA Journal, Volume 26, pp. 612–618.

Bert, C. W. and Malik, M., 1996. Differential quadrature method in computational mechanics: A review. Applied Mechanics Reviews, Volume 49, pp. 1–28.

Beslin, O. and Nicolas, J., 1997. A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions. Journal of Sound and Vibration, 202(5), pp. 633–655.

Bokaian, A., 1988. Natural frequencies of beams under compressive axial load. Journal of Sound and Vibration, Volume 126, pp. 49–65.

Boukhalfa, A. and Hadjoui, A., 2014. Dynamic analysis of a spinning functionally graded material shaft by the p–version of the finite element method. Latin American Journal of Solids and Structures.

Chen and New, C., 1999. A differential quadrature finite element method. Applied mechanics in the Americas. Proceedings of the 6th Pan-American Congress of Applied Mechanics and 8th International Conference on Dynamic Problems in Mechanics PACAM VI, Rio de Janeiro, Brazil; US, pp. 305–308.

Choi, I. S., Pierre, C. and Ulsoy, A. G., 1992. Consistent modeling of rotating Timoshenko shaft subject to axial loads. Journal of Vibration and Acoustics, Volume 114, pp. 249–259.

Clough, R. W., 1960. The finite element method in plane stress analysis Conference on Electronic Computation, Pittsburgh, PA. Proceedings of the Second American Society of Civil Engineers, pp. 345–378.

Cuiyun, L. et al., 2016. A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains. International Journal for Numerical Methods in Engineering, 109(2), pp. 174–197.

Dakel, M., 2013. Steady-state dynamic behaviour of an on-board rotor under combined base motions. Journal of Vibration and Control, Volume 20, pp. 2254–2287.

Duchemin, M., Berlioz, A. and Ferraris, G., 2006. Dynamic behavior and stability of a rotor under base excitation. Journal of Vibration and Acoustics, Volume 128, pp. 576–585.

Eshlenman, R. L. and Eubanks, R. A., 1967. On the critical speeds of a continuous shaft-disk system. Journal of Engineering for Industry, Volume 80, pp. 645–652.

Fellah, A., Hadjoui, A. and Bekhaled, B. S. A., 2019. Study of the Effect of an Open Transverse Crack on the Vibratory Behavior of Rotors Using the hp Version of the Finite Element Method. Journal of Solid Mechanics, 11(1), pp. 181–200.

Green, R. B., 1948. Gyroscopic effects on the critical speeds of flexible rotors,. Transaction of the American society of Mechanic Engineers., Volume 70, pp. 309–376.

Hassan, A., Abdelhamid, H. and Saimi, A., 2020. Numerical analysis on the dynamics behavior of FGM rotor in thermal environment using hp finite element method. Mechanics Based Design of Structures and Machines, pp. 1–24.

Heuveline, V. and Rannacher, R., 2003. Duality-Based Adaptivity in the Hp-Finite Element Method. Journal of Numerical Mathematics, 11(2), pp. 95–113.

Malik, M. and Bert, C., 2000. Vibration analysis of plates with curvilinear quadrilateral planforms by DQM using blending functions. Journal of Sound and Vibration, 230(4), pp. 949–954.

Nelson, H. D. and McVaugh, J. M., 1976. The dynamics of rotor-bearing systems using finite elements. ASME Journal of Engineering for Industry, Volume 98, pp. 593–600.

O. C. Zienkiewicz, M. L., 1977. The finite element method, 3rd edn. Wiley Online Library, pp. 1054–1054.

Oden, J. T. a. D. L., 1991. h-p adaptive finite element methods in computational fluid dynamics. Computer Methods in Applied Mechanics and Engineering, Volume 89, pp. 11–40.

Peano, A., 1976. Hierarchies of conforming finite elements for plane elasticity and plate bending.. Computers & Mathematics with Applications, Volume 2, pp. 211–224.

Petyt, M., 2010. Introduction to Finite Element Vibration Analysis (2nd edn). New York: Cambridge University Press.

Rao, S. S., 1983. Rotor dynamics. New York: NY: Wiley.

René, J. G., 1988. Vibrations des structures : interactions avec les fluides, sources d’excitation aléatoires. Paris: Eyrolles.

Ri, K. et al., 2020. Nonlinear forced vibration analysis of composite beam combined with DQFEM and IHB. AIP Advances, 10(8).

Saimi, A. H. A., 2016. An engineering application of the h-p version of the finite elements method to the dynamics analysis of a symmetrical on-board rotor. European Journal of Computational Mechanics, pp. 388–416.

Sajal, S. S. et al., 2018. Dynamic analysis of microbeams based on modified strain gradient theory using differential quadrature method. European Journal of Computational Mechanics , 27(3), pp. 187–203.

Shu, C., 2000. Differential Quadrature and its Application in Engineering. Springer-Verlag d. London: s.n.

Striz, A. G., Chen, W. L. and Bert, C. W., 1995. High accuracy plane stress and plate elements in the quadrature element method. Proceedings of the 36th AIAA/ASME/ASCE/AHS/ASC, pp. 957–965.

Striz, A. G., Chen, W. L. and Bert, C. W., 1997. Free vibration of plates by the high accuracy quadrature element method. Journal of Sound and Vibration, Volume 202, pp. 689–702.

Suri, M., 1997. A reduced constraint h?

?

finite element method for shell problems. Mathematics of computation, 66(217), pp. 15–29.

Szabo, B., 1979. Some recent developments in finite element analysis. Computers & Mathematics with Applications , 5(2), pp. 99–115.

Szabo, B. and Prob, 1985. Theoretical manual release 1. St Louis, Missouri: Noetic Technologies Corporation.

Weiyan, Z., Feng, G. and Yongsheng, R., 2019. Generalized Differential Quadrature Method for Free Vibration Analysis of a Rotating Composite Thin-Walled Shaft. Mathematical Problems in Engineering, Volume 2019.

Xing, Y. F. and Liu, B., 2009. High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. International Journal for Numirical methods in engineering. Vol 80. issue 13. pp. 1718–1742.

Yufeng, X., Bo, L. and Guang, L., 2010. A Differential Quadrature Finite Element Method. International Journal of Applied Mechanics, 2(1), pp. 207–227.

Zhong, H. and Yu, T., 2009. A weak form quadrature element method for plane elasticity problems. Applied Mathematical Modelling, 33(10), pp. 3801–3814.

Zienkiewicz, O., De, S., Gago, J. and Kelly, D., 1983. The hierarchical concept in finite element analysis. Computers & Structures, 16(1–4), pp. 53–65.

Zienkiewicz, O., Irons, B., Scott, F. and Campbell, J., 1970. Three-dimensional stress analysis. Proceedings of IUTAM Symposium on High Speed Computing of Elastic Structures, Liege, pp. 413–431.