Internal resonances in a transversally excited imperfect circular cylindrical shell

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Available online at www.sciencedirect.com Procedia Engineering00 (2017) 000–000

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Procedia Engineering 199 (2017) 838–843

X International Conference on Structural Dynamics, EURODYN 2017

Internal resonances in a transversally excited imperfect circular cylindrical shell Lara Rodriguesa, Paulo B. Gonçalvesa, Frederico M. A. Silvab,* a Pontifical

Pontifical Catholic University of Rio de Janeiro, Rua Marquês de São Vicente no. 255, Rio de Janeiro, 22451-900, Brazil b Federal University of Goiás, Av. Universitária no. 1488, Goiânia, 74605-200, Brazil

Abstract Cylindrical shells exhibit a dense frequency spectrum, especially near the lowest frequency range. In addition, due to the circumferential symmetry, frequencies occur in pairs. So, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to a complex dynamic behavior. The aim of the present work is to investigate the influence of several modal geometrical imperfections on the nonlinear vibration of simply supported transversally excited cylindrical shells with multiple equal or nearly equal natural frequencies. The shell is modelled using the Donnell nonlinear shallow shell theory and the discretized equations of motion are obtained by applying the Galerkin method. For this, a modal solution that takes into account the modal interaction among the relevant modes and the influence of their companion modes (modes with rotational symmetry), which satisfies the boundary and continuity conditions of the shell, is derived. Several numerical strategies are used to study the nonlinear behavior of the imperfect shell. Special attention is given to the shape and the magnitude of the initial geometric imperfection on the resonance curves and bifurcations of simply supported transversally excited cylindrical shells with 1:1:1:1 internal resonance (four interacting modes). © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility ofthe organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords:internal resonances; geometrical imperfections; nonlinear analysis; cylindrical shells.

1. Introduction The development of consistent modal solutions capable of describing the main modal couplings and interactions observed in cylindrical shells has received much attention in literature. A detailed review of this subject can be found

* Corresponding author. Tel.: +55-62-3209-6179; fax: +55-62-3209-6084. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility ofthe organizing committee of EURODYN 2017.

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.010

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in [1-3]. The consideration of these phenomena has also shown to be essential in correctly describing the nonlinear vibrations of cylindrical shells and detect possible instabilities and coexisting coupled and uncoupled solutions. Due to the circumferential symmetry, in shells of revolution frequencies occur in pairs. Also, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to a strong nonlinear interaction among these modes under dynamic conditions, influencing thereby the response and safety of the shell and changing considerably its stability boundaries and bifurcation diagrams [6, 7]. However, no consistent modal solution taking into account the simultaneous effect of modal coupling plus modal interaction for imperfect cylindrical shells is found in literature [8, 9]. To understand the nonlinear vibrations of cylindrical shells, the investigation of the effects of initial geometric imperfections is crucial. Geometric imperfections are known to reduce the load carrying capacity of cylindrical shells under static loads, being empirical knock down factors prescribed by several international standards as, for example, [4, 5], to obtain a safe design load. The influence of geometric imperfections on the response of axially or transversally exciting empty and fluid-loaded cylindrical shells has been studied by several authors in recent years [1-3]. However, the influence of the imperfections shape and magnitude on the nonlinear vibrations and bifurcation diagrams in the light of the nonlinear modal coupling is not found in literature. So, the aim of this work is to study the nonlinear behavior of the shell considering different shapes of initial geometric imperfections on the resonance curves of simply supported transversally excited cylindrical shells, considering the internal resonances of type 1:1:1:1. For this, a consistent modal solution, based on previous works [10], is derived. Special attention is dedicated to initial geometric imperfections that are considered in the form of the one of the fundamental vibration modes used as the initial solution. The 1:1:1:2 internal resonances of a fluid-filled shell were previously studied in [11]. 2. Formulation of the problem. 2.1. Shell equations Consider a simply supported cylindrical shell with radius R, thickness h, length L and an initial geometric imperfection described by a function wi, made with an elastic material with Young’s modulus E, Poisson coefficient  and mass density . The geometry, coordinate system (x, , z) and displacements (u, v, w) are shown in Fig. 1. The nonlinear equation of motion based on the Von Kármán-Donnell shallow shell theory, in terms of the transversal displacements and a stress function, are given by:

   1  h  0 w   2 D  4 w  D 4 w  p (t )   hw 2

F, x R2

4F 

w  wi , x Eh R

4

 w  wi , 1   F, xx    R R2 

F, R2

w  wi , xx

( w,2x  w, xx w,  R w, xx  2w, x wi , x  w, xx wi ,  w, wi , xx )

(1.1)

(1.2)

where 0 is the natural frequency of the cylindrical shell, 1 and 2 are, respectively, the linear viscous damping and the viscoelastic material damping coefficients and F is the stress function. The shell is subjected to a time dependent harmonic lateral pressure that excites de fundamental vibration mode defined by:

 m x  p  t  PL sin   cos n cos  L t   L 

(2)

Lara Rodrigues et al. / Procedia Engineering 199 (2017) 838–843 Lara Rodrigues, Paulo B. Gonçalves, Frederico M. A. Silva / Procedia Engineering 00 (2017) 000–000

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3

where m and n are, respectively, the number of longitudinal half-waves and the number of circumferential wave; L is the excitation frequency and PL is the magnitude of the lateral pressure.

Fig. 1. Geometry, coordinate system and displacement field of the shell.

Fig. 2. Nondimensional natural frequencies for a perfect cylindrical shell, considering different values for L and n. (m = 1).

The following non-dimensional parameters are adopted in the parametric analysis: W

w h

Wi 

wi h



x L

  0 P t

L 

L  0P

 



0 R 2  1 2



(3)

E

where 0P is the lowest natural frequency of the perfect cylindrical shell while 0 is the lowest natural frequency of each studied case. 2.2. General solution for axial and circumferential displacement fields as a function of transversal displacement As an example, consider a simply supported cylindrical shell with radius R = 0.2 m, L = 0.23081 m and h = 0.002 m. The shell material is homogeneous and isotropic with Young´s modulus E = 210 GPa, Poisson's ratio  = 0.3 and density  = 7850 kg/m³. The vibration modes of a simply supported cylindrical shells exhibit m half-waves in the axial direction and n waves in the circumferential direction. For several geometries the shell has two vibration modes with the same natural frequency, leading to an internal resonance and a complex nonlinear behavior under transversal excitation [10]. For the present geometry, the values of the lowest nondimensional natural frequencies, ’, for modes (m, n) = (1, 6) and (m, N) = (1, 7) are 0.20548 as shown in Fig. 2 and marked by . So, this cylindrical shell has two distinct vibration modes with the same natural frequency. In addition, due to the circumferential symmetry, frequencies occur in pairs. This leads to a 1:1:1:1 internal resonance. For the application of perturbation techniques, the analysis must begin by considering these four interacting modes as the modal initial solution (seed modes) in the perturbation procedure, i.e. [10]:





W  A111  cos ( n )  B111  sin ( n )  A 2 11  cos ( N  )  B 2 11  sin ( N  ) sin m   

(4)

where N = n + 1 and A111(), A211(), B111() and B211() are the time dependent modal amplitudes of linear modes. By applying the perturbation procedures, as described in [10], and by considering the boundary conditions for a simply supported cylindrical shell,

w0,  wL,  0 and M x 0,  M x L,  0

(5)

the following modal solution that accounts for the modal coupling and interaction is obtained for the transversal displacement:

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Rodrigues et al. /M. Procedia 199 (2017) 838–843 Lara Rodrigues, Paulo B.Lara Gonçalves, Frederico A. SilvaEngineering / Procedia Engineering 00 (2017) 000–000

W

841

  A1ij ( ) cos ( in )  B1ij ( ) sin( in )  A2ij ( ) cos ( iN )  B2ij ( ) sin( iN ) sin  j m    N1



i 1, 3, 5 j 1, 3, 5



   A3ij ( ) cos  i N   N1  i n   A4ij ( ) cos  i N   N1  i n 

N1  1



i 1, 2, 3 j 1, 3, 5

 



  

 

 B3ij ( ) sin i N  N1  i n   B 4ij ( ) sin i N  N1  i n  sin  j m    

N2



 

  0, 2, 4   0

A5 

 2  6   ( ) cos

  n   B5 2 6  ( ) sin n 

 3  6   A6 2  6   ( ) cos   N    B6 2  6   ( ) sin N    cos 6  m     4  12  1 6  cos  4  6  m     cos  2  6   m      4 12  





   A7 2 6  ( ) cos   N   N 2  n     B7 2 6  ( ) sin   N   N 2  n   

N 2 1



 1, 2, 3   0





 





   

 A8 2  6   ( ) cos  N  N 2   n   B8 2  6   ( ) sin  N  N 2   n 

 3  6  1 6  cos  4  6  m    cos 6  m     cos  2  6   m          4 12  4 12   

(6)

The additional terms in Eq. (6) are due to the quadratic and cubic nonlinearities in the equations of motion. By retaining in Eq. (6) a sufficiently large number of terms to achieve convergence up to large vibration amplitudes [10], substituting this approximate expansion into Eq. (1.2) and solving analytically the partial differential equation, a consistent solution for the stress function F is obtained. Finally, by substituting the adopted expansion for the transversal displacement together with the obtained stress function into Eq. (1.1) and by applying the standard Galerkin method, a discretized system of ordinary differential equations of motion in time domain is derived. 3. Numerical results To study the nonlinear behavior of the shell considering the influence of initial geometric imperfections, a length L = 0.23081 m is here adopted. For this value of L, 1:1:1:1 internal resonance between the linear modes (m, n)=(1, 6) and (m, N)=(1, 7) occurs. In this work, the following four different shapes of initial geometric imperfections are considered in the analyses: Wi W11i cos n sin m x L  ; Wi W11i sin n sin m x L 

Wi W11i cos N  sin m x L  ; Wi W11i sin N  sin m x L 

(7-8) (9-10)

where W11i are the non-dimensional amplitude of the initial geometric imperfection. Friedrich et al. [12] suggest, as a guidance, when choosing an imperfection magnitude, the expression W11i  0.025 (  2.8 

R / h ) , which can be

used for preliminary design purposes. This leads to a knock down factor similar to those proposed by some international codes such as NASA SP-8007 [4] Eurocode 3 [5] for shells under static loading. To achieve the convergence up to large vibration amplitudes of the order of the shell thickness, the modes A111(), B111(), A211(), B211(), A502(), A522(), B522(), A622(), B622(), A712(), B712(), A812(), B812(), A113(), B113(), A213(), B213(), A131(), B131(), A231(), B231(), A311(), B311(), A411(), B411(), A321(),

Lara Rodrigues et al. / Procedia Engineering 199 (2017) 838–843 Lara Rodrigues, Paulo B. Gonçalves, Frederico M. A. Silva / Procedia Engineering 00 (2017) 000–000

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5

B321(), A421() and B421() are selected in Eq. (6), leading to a 29-dof system where the modal amplitudes in bold are due to the modal interaction [9].

(a)

(b)

(c)

(d)

Fig. 3. Resonance curves of imperfect cylindrical with different values of initial geometrical imperfection; (a) and (b) in the form of driven mode with n=6 waves – Eq. (7) and (c) and (d) in the form of companion mode with n=6 waves – Eq. (8). PL = 600 N/m², 1 = 0.001, 2 = 0 and m = 1.

Fig. 3(a) and 3(b) show the variation of the modal amplitudes of the driven and companion modes, respectively, as a function of the forcing frequency for selected magnitudes of the imperfection in the form the driven mode, Eq. (7). The resonance curves moves to the left as W11i increases due to a decrease in the natural frequencies [13]. For small imperfection in the form of driven mode, two bifurcations occur along the resonant branch in Fig. 3(a). They disappear as the imperfection increases (see curves for W11i > 0.18) and the companion mode, Fig. 3(b), is not excited. Fig. 3(c) and 3(d) show the results for an imperfection in the form of the companion mode, Eq. (8). Here, the two bifurcations along the resonance curve remain, independent of the value of the imperfections amplitude, but its stability changes from stable to unstable for high value of imperfections amplitude. As a consequence, the companion mode is always excited. In both cases, there is no modal interaction between the modes (m, n)=(1, 6) and (m, N)=(1, 7), in spite of the fact that, for the perfect shell, both have the same natural frequency.

(a)

(b)

(c)

(d)

Fig. 4. Resonance curves of imperfect cylindrical with different values of initial geometrical imperfection in form of driven mode with N=7 waves – Eq. (9). PL = 600 N/m², 1 = 0.001, 2 = 0 and m = 1.

When the initial geometric imperfection assume the form of the driven or the companion mode, with N=7 circumferential waves, Eq. (9) and Eq. (10) respectively, some important modifications are identified, as shown in Figs. 4 and 5. In these cases, the modal interaction between the modes (m, n)=(1, 6) and (m, N)=(1, 7), as illustrated by the Figs. 4 (c), 4 (d), 5 (c) and 5 (d), occurs, being all modes in Eq. (6) excited, independent of the magnitude of the geometric imperfection, although the applied external pressure excite only the driven mode (m, n). Again, all curves move to the left as the imperfection magnitude increases. Several bifurcations are observed along the resonance curves, as illustrated in Figs. 4(a) and 5(a), being its location and the ensuing secondary path highly dependent on the imperfection magnitude. This collection of results shows that geometric imperfections have a noticeable influence on the nonlinear vibrations of cylindrical shells and, as observed here, on possible modal couplings and interactions. It must be pointed out that the imperfections here used are small (just a fraction of the shell thickness) and within the limits of those found in high precision manufacturing processes. One should also bear in mind that actual geometric imperfections may have important modal components in all considered modes. Around the resonance, the response is chaotic when the travelling mode vibration is activated beyond a threshold of vibration amplitude, which is low [14]. In all examples the maximum vibration amplitudes is much lower than the shell thickness, within the validity of the adopted shell theory and the convergence range of the present modal expansion.

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Lara Rodrigues et al. / Procedia Engineering 199 (2017) 838–843 Lara Rodrigues, Paulo B. Gonçalves, Frederico M. A. Silva / Procedia Engineering 00 (2017) 000–000

(a)

(b)

(c)

843

(d)

Fig. 5. Resonance curves of imperfect cylindrical with different values of initial geometrical imperfection in form of companion mode with N=7 waves – Eq. (10). PL = 600 N/m², 1 = 0.001, 2 = 0 and m = 1.

4. Conclusions This paper investigates the influence of the shape and magnitude of small initial geometric imperfections on the nonlinear vibrations and stability of an imperfect cylindrical shell. Important modifications in the nonlinear forced vibration can be observed. In all cases analyzed here, the shell displays a softening nonlinear behavior. The results show that energy transfer between the directly excited mode (driven mode) to the companion mode and other modes with equal or nearly equal natural frequencies depends on the shape and magnitude of the initial geometric imperfection, influencing consequently the stability of the resonant branch. This work is a work in progress. Future work will consider the influence of the initial imperfection in the shape of the nonlinear vibration mode, axisymmetric modes and arbitrary imperfections on the results. Acknowledgements This work was made possible by the support of the Brazilian Ministry of Education, CAPES, CNPq, FAPERJCNE and FAPEG. References [1] V. D.Kubenko, P. S.Koval'chuk, Nonlinear problems of the vibration of thin shells (review), International Applied Mechanics 34 (1998) 703728. [2] M. Amabili,M. P. Païdoussis, Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction, Applied Mechanics Reviews 56 (2003) 655-699. [3] F. Alijani, M.Amabili, Non-linear vibrations of shells: A literature review from 2003 to 2013, International Journal of Non-Linear Mechanics 58 (2014) 233-257. [4] NASASP-8007,Buckling of thin-walled circular cylinders, NASA space vehicle design criteria, 1968. [5] EN 1993-1-6, Eurocode3: design of steel structures – Part 1–6: Strength and stability of shell structures, English version, 2007. [6] P. B. Gonçalves, R. C. Batista, Non-linear vibration analysis of fluid-filled cylindrical shells, Journal of Sound and Vibration 127 (1988) 133143. [7] A. A. Popov, J. M. T. Thompson, F. A. McRobie, Low dimensional models of shell vibrations: parametrically excited vibrations of cylindrical shells. Journal of Sound andVibration 209 (1998) 163-186. [8] P. B. Gonçalves, Z. J. G. N. Del Prado, Effect of nonlinear modal interaction on the dynamic instability of axially excited cylindrical shells. Computers and Structures 82 (2004) 2621–2634. [9] M. Amabili, Internal resonances in non-linear vibrations of laminated circular cylindrical shell, Nonlinear Dynamics 69 (2012) 755-770. [10] L. Rodrigues, F. M. A. Silva, P. B. Gonçalves, Z. J. G. N. Del Prado, Effects of modal coupling on the dynamics of parametrically and directly excited cylindrical shells, Thin-Walled Structures 81 (2013) 210-224. [11] M. Amabili, F. Pellicano and A. F. Vakakis, Nonlinear vibrations and multiple resonances of fluid-filled, circular shells, Part I: Equations of motion and numerical results, Transactions of the ASME, Journal of Vibration and Acoustics 122 (2000) 346-354. [12] L. Friedrich, T-A. Schmid-Fuertes, K-U. Schröder, Comparison of theoretical approaches to account for geometric imperfections of unstiffened isotropic thin walled cylindrical shell structures under axial compression, Thin-Walled Structures 92 (2015) 1-9. [13] L. Rodrigues P. B. Gonçalves, F.M.A Silva, The influence of modal geometrical imperfections on the nonlinear vibrations of a thin-walled circular cylindrical shell, ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 8, 28th Conference on Mechanical Vibration and Noise, V008T10A018, doi:10.1115/DETC2016-59649. [14] M. Amabili, P. Balasubramanian, G. Ferrari, Travelling wave and non-stationary response in nonlinear vibrations of circular cylindrical shells: experiments and simulations, Journal of Sound and Vibration 381 (2016) 220-245.