# A hierarchical generalized formulation for the large-displacement dynamic analysis of rotating plates

## Abstract

A hierarchical formulation is presented for the large displacement modal and transient analysis of rotating plates having different thicknesses, sizes, and settings angle with respect to the spinning axis. After derivation of the governing equations of motion by the Principle of Virtual Displacements, the finite element discretization yields a set of nonlinear ordinary-differential equations for the generalized coordinates including gyroscopic terms, centrifugal/Euler accelerations and spin-softening effects. A Total Lagrangian approach is adopted. The modal analysis is performed by a two-step procedure. The static shape of the structure deformed by the centrifugal force is first defined. Subsequently, eigenfrequencies and mode shapes are computed by a classical eigenvalue problem past the static configuration previously computed. The transient analysis of the plates subject to a varying angular velocity is tackled by solving the nonlinear equations of motion by the Generalized-$$\alpha$$ method coupled to the Newmark’s approximation for the velocity and acceleration fields. It has been numerically proven that accurate results can be obtained by the use of Murakami’s Zig-Zag Function which a-priori enforces the interlaminar discontinuity of the displacements’slopes in the Equivalent Single Layer axiomatic model, thus avoiding higher-order polynomial representations for the displacement fields, or the use of Layer-Wise theories.

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1. 1.

For conciseness, a mixed variant-invariant (vectors-arrays) notation is used in the following

2. 2.

If a structural damping is modeled with a matrix $$\varvec{C}_{\mathrm {STR}}$$ then an additional term of the type $$\varvec{C}_{\mathrm {STR}} \cdot \dot{\varvec{U}}$$ appears in the equations of motion.

3. 3.

In the practical programming the inversion of the matrix is avoided by using different method of calculating the eigenvalues; the inversion is showed here to just represent the eigenvalue problem in canonic form.

## References

1. 1.

NX Nastran user’s guide. Siemens Product Lifecycle Management Software Inc. (2014)

2. 2.

COMSOL multiphysics v. 5.4. www.comsol.com, COMSOL AB, Stockholm, Sweden (2018)

3. 3.

Aitharaju VR, Averill RC (1999) $$c^0$$ zig-zag finite element for analysis of laminated composite beams. J Eng Mech 125(3):323–330

4. 4.

Argyris JH, Mlejnek HP (1991) Dynamics of structures. Texts on computational mechanics. North-Holland, Amsterdam

5. 5.

Averill RC, Yip YC (1996) Thick beam theory and finite element model with zig-zag sublaminate approximations. AIAA J 34(8):1627–1632

6. 6.

Babu AA, Vasudevan R (2017) Vibration analysis of rotating delaminated non-uniform composite plates. Aerosp Sci Technol 60:172–182

7. 7.

Banerjee AK, Kane TR (1989) Dynamics of a plate in large overall motion. J Appl Mech 56:887–892

8. 8.

Bhumbla R, Kosmatka JB (1996) Behavior of spinning pretwisted composite plates using a nonlinear finite element approach. AIAA J 34(8):1686–1695

9. 9.

Carne T, Lobitz D, Nord A, Watson R (1981) Finite element analysis and modal testing of a rotating wind turbine, p 696

10. 10.

Carrera E, Cinefra M, Petrolo M, Zappino E (2014) Finite element analysis of structures through unified formulation. Wiley, New York

11. 11.

Carrera E, Demasi L (2002) Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: derivation of finite element matrices. Int J Numer Methods Eng 55(2):191–231

12. 12.

Carrera E, Demasi L (2002) Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: numerical implementations. Int J Numer Methods Eng 55(3):253–291

13. 13.

Castillo Pardo A, Goulos I, Pachidis V (2017) Modelling and analysis of coupled flap-lag-torsion vibration characteristics helicopter rotor blades. Proc Inst Mech Eng Part G J Aerosp Eng 231(10):1804–1823

14. 14.

Chen Y, Zhang D, Li L (2019) Dynamics analysis of a rotating plate with a setting angle by using the absolute nodal coordinate formulation. Eur J Mech-A/Solids 74:257–271

15. 15.

Demasi L (2008) $$\infty ^3$$ hierarchy plate theories for thick and thin composite plates: the generalized unified formulation. Compos Struct 87(3):256–270

16. 16.

Demasi L (2009) $$infty ^6$$ mixed plate theories based on the generalized unified formulation. Part V: results. Compos Struct 84:195–205

17. 17.

Demasi L, Ashenafi Y, Cavallaro R, Santarpia E (2015) Generalized unified formulation shell element for functionally graded variable-stiffness composite laminates and aeroelastic applications. Compos Struct 131:501–515

18. 18.

Demasi L, Biagini G, Vannucci F, Santarpia E, Cavallaro R (2017) Equivalent single layer, zig-zag, and layer wise theories for variable angle tow composites based on the generalized unified formulation. Compos Struct 177:54–79

19. 19.

Dokainish MA, Rawtani S (1971) Vibration analysis of rotating cantilever plates. Int J Numer Methods Eng 3(2):233–248

20. 20.

Du CF, Zhang DG, Liu GR (2019) A cell-based smoothed finite element method for free vibration analysis of a rotating plate. Int J Comput Methods 16(05):1840003

21. 21.

Fang JS, Zhou D (2017) Free vibration analysis of rotating mindlin plates with variable thickness. Int J Struct Stab Dyn 17(04):1750046

22. 22.

Farhadi S, Hosseini-Hashemi SH (2011) Aeroelastic behavior of cantilevered rotating rectangular plates. Int J Mech Sci 53(4):316–328

23. 23.

Filippi M, Pagani A, Carrera E (2018) Accurate nonlinear dynamics and mode abberration of rotating blades. J Appl Mech Trans 85:2–8

24. 24.

Filippi M, Pagani A, Carrera E (2019) Three-dimensional solutions for rotor blades using high-order geometrical nonlinear beam finite elements. J Am Helicopter Soc 64:1–10

25. 25.

Hashemi SH, Farhadi S, Carra S (2009) Free vibration analysis of rotating thick plates. J Sound Vib 323:366–384

26. 26.

Hashemi SM, Richard MJ (2001) Natural frequencies of rotating uniform beams with coriolis effects. J Vib Acoust 123(4):444–455

27. 27.

Hodges DH, Dowell EH (1974) Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA/TN-7818

28. 28.

Hodges DH, Ormiston RA (1976) Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling. NASA/TN-8192

29. 29.

Hu XX, Sakiyama T, Matsuda H, Morita C (2004) Fundamental vibration of rotating cantilever blades with pre-twist. J Sound Vib 271(1–2):47–66

30. 30.

Jinyang L, Jiazhen H (2005) Geometric nonlinear formulation and discretization method for a rectangular plate undergoing large overall motions. Mech Res Commun 32(5):561–571

31. 31.

Júnior CJF, Cardozo ACP, Júnior VM, Neto AG (2019) Modeling wind turbine blades by geometrically-exact beam and shell elements: a comparative approach. Eng Struct 180:357–378

32. 32.

Kapuria S, Dumir PC, Ahmed A (2003) An efficient higher order zigzag theory for composite and sandwich beams subjected to thermal loading. Int J Solids Struct 40(24):6613–6631

33. 33.

Kar RC, Sujata T (1992) Dynamic stability of a rotating, pretwisted and preconed cantilever beam including coriolis effects. Comput Struct 42(5):741–750

34. 34.

Karmakar A, Sinha PK (1997) Finite element free vibration analysis of rotating laminated composite pretwisted cantilever plates. J Reinf Plast Compos 16(16):1461–1491

35. 35.

Kim SK, Yoo HH (2002) Vibration analysis of rotating composite cantilever plates. KSME Int J 16(3):320–326

36. 36.

Laurenson RM (1976) Modal analysis of rotating flexible structures. AIAA J 14(10):1444–1450

37. 37.

Li L, Zhang DG (2016) Free vibration analysis of rotating functionally graded rectangular plates. Compos Struct 136:493–504

38. 38.

Likins PW (1972) Finite element appendage equations for hybrid coordinate dynamic analysis. Int J Solids Struct 8(5):709–731

39. 39.

Liu L, Zhang Z, Hua H (2007) Dynamic characteristics of rotating cantilever plates with active constrained layer damping treatments. Smart Mater Struct 16(5):1849

40. 40.

Meirovitch L (1974) A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA J 12(10):1337–1342

41. 41.

Meirovitch L (2010) Methods of analytical dynamics. Courier Corporation

42. 42.

Murakami H (1986) Laminated composite plate theory with improved in-plane responses. J Appl Mech 53:661–666

43. 43.

Rao JS, Gupta K (1987) Free vibrations of rotating small aspect ratio pretwisted blades. Mech Mach Theory 22(2):159-167

44. 44.

Rostami H, Ranji AR, Bakhtiari-Nejad F (2016) Free in-plane vibration analysis of rotating rectangular orthotropic cantilever plates. Int J Mech Sci 115:438–456

45. 45.

Rostami H, Ranji AR, Bakhtiari-Nejad F (2018) Vibration characteristics of rotating orthotropic cantilever plates using analytical approaches: a comprehensive parametric study. Arch Appl Mech 88(4):481–502

46. 46.

Santarpia E (2020) A variable kinematic model for large deection of functionally graded variable-stiness composite laminates. PhD thesis, University of California San Diego and San Diego State University

47. 47.

Santarpia E, Demasi L (2020) Large displacement models for composites based on Murakamihc’s zig-zag function, Green–Lagrange strain tensor, and generalized unified formulation. Thin-Walled Struct 150:106460

48. 48.

Shabana AA, Christensen AP (1997) Three-dimensional absolute nodal co-ordinate formulation: plate problem. Int J Numer Methods Eng 40(15):2775–2790

49. 49.

Sinha SK, Turner KE (2011) Natural frequencies of a pre-twisted blade in a centrifugal force field. J Sound Vib 330(11):2655–2681

50. 50.

Sinha SK, Zylka RP (2017) Vibration analysis of composite airfoil blade using orthotropic thin shell bending theory. Int J Mech Sci 121:90–105

51. 51.

Subrahmanyam KB, Kaza KRV, Brown GV, Lawrence C (1987) Nonlinear vibration and stability of rotating, pretwisted, preconed blades including coriolis effects. J Aircr 24(5):342–352

52. 52.

Tisseur F, Meerbergen K (2001) The quadratic eigenvalue problem. SIAM Rev 43(2):235–286

53. 53.

Tornabene F, Fantuzzi N, Bacciocchi M (2019) Refined shear deformation theories for laminated composite arches and beams with variable thickness: natural frequency analysis. Eng Anal Boundary Elem 100:24–47

54. 54.

Yoo HH, Chung J (2001) Dynamics of rectangular plates undergoing prescribed overall motion. J Sound Vib 239(1):123–137

55. 55.

Yoo HH, Kim SK, Inman DJ (2002) Modal analysis of rotating composite cantilever plates. J Sound Vib 258(2):233–246

56. 56.

Yoo HH, Pierre C (2003) Modal characteristic of a rotating rectangular cantilever plate. J Sound Vib 259(1):81–96

57. 57.

Zhao J, Tian Q, Hu H (2011) Modal analysis of a rotating thin plate via absolute nodal coordinate formulation. J Comput Nonlinear Dyn 6(4):041013

## Acknowledgements

The authors wish to thank Prof. H.H.Yoo of the Hanyang University—Department of Mechanical Engineering—for having shared some numerical results used throughout the paper for validation purposes.

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Correspondence to Claudio Testa.

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## A GUF matrices

### Centrifugal forces

The centrifugal acceleration in a finite element formulation generates both an additional stiffness matrix and an apparent force vector. Its contribution to the virtual work is: (96) (97)

Performing the spatial discretization of : (98)

Operating with a similar approach, is written as (99)

The stiffness and force kernels of the apparent centrifugal force due to the rotation of the body can be identified: (100)

For example: (101) (102)

### Coriolis forces

The Coriolis acceleration provides an additional damping matrix added, if present, to the structural ones. The virtual work contribution is: (103)

GUF as well the spatial discretization are applied to both the virtual displacement and the velocity. e.g.

\begin{aligned}&\,_0\delta u_x(x ,y ,z) = \;{}^x_0F_{\alpha _{u_x} } (z) \;{}^x_0N_I(x,y)\; \,_0\delta U_{x\alpha _{u_x}I } (x ,y) \end{aligned}
(104)
\begin{aligned}&\,_0\dot{u}_x(x ,y ,z) \;= \;{}^x_0F_{\alpha _{u_x} } (z) \;{}^x_0N_I(x,y)\; \,_0\dot{U}_{x\alpha _{u_x}I } \;\, (x ,y) \end{aligned}
(105)

where

\begin{aligned} \begin{array}{l} \alpha _{u_x} \!\! = t,l,b \quad l = 2,\dots ,N_{u_x} \quad I = 1,2,\dots M_n \end{array} \end{aligned}
(106)

and $$M_n$$ is the number of nodes of the element.

The explicit expression for the semi-discretized form of Eq. 103 is (107)

It is possible to recognize the gyroscopic damping matrix: (108)

where for example the kernel of the matrix relating the $$\alpha _{ux}$$ term of the polynomial/Legendre function for the x displacement of the node I and the $$\alpha _{uy}$$ for the y velocity function of node J is (109)

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