Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells

Transcription

1 Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells Abstract The nonlinear vibration and noral ode shapes of FG cylindrical shells are investigated using an efficient analytical ethod. The equations of otion of the shell are based on the Donnell s nonlinear shallow-shell, and the aterial is assued to be gradually changed across the thickness according to the siple power law. The solution is provided by first discretizing the equations of otion using the ulti-ode Galerkin s ethod. The nonlinear noral ode of the syste is then extracted using the invariant anifold approach and eployed to decouple the discretized equations. The hootopy analysis ethod is finally used to deterine the nonlinear frequency. Nuerical results are presented for the backbone curves of FG cylindrical shells, nonlinear ode shapes and also the nonlinear invariant odal surfaces. The volue fraction index and the geoetric properties of the shell are found to be effective on the type of nonlinear behavior and also the nonlinear ode shapes of the shell. The circuferential half-wave nubers of the nonlinear ode shapes are found to change with tie especially in a thinner cylinder. Saeed Mahoudkhani a a Aerospace Engineering Departent, Faculty of New Technologies and Engineering, Shahid Beheshti University, GC, Velenjak Sq., Tehran, Iran, s_ahoudkhani@sbu.ac.ir Received.8.6 In revised for 7..6 Accepted 6..6 Available online 9..7 Keywords FG cylindrical shells, Invariant anifold (IM), nonlinear noral ode (NNM), Hootopy analysis ethod (HAM), nonlinear ode shapes INTRODUCTION Functionally graded aterials (FGM s) are the new types of coposite aterials that have the unique feature of both high theral and high echanical properties. The constituent aterials of FGMs (usually ceraic and etal) are cobined such that the variation of their volue fractions is continuous, giving rise to sooth and gradual change of the echanical and theral properties. Hence in spite of the traditional coposites that ay suffer high aounts of theral stresses at the interfaces, FGMs could withstand high teperatures while aintaining their structural integrities and providing desirable echanical properties. These properties akes FGMs a suitable choice for

2 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells use in aerospace structures such as outer skins of high speed vehicles, which are coonly in the for of cylindrical shells. The analysis of functionally graded (FG) cylindrical shells is therefore of critical iportance and could lead to ore efficient structures in the future. Vibration analyses are especially one of the ain and critical considerations and thus should be given a high priority in designing cylindrical shell used in aerospace application. Nuerous studies have been perfored so far on the linear vibration of FG cylindrical shell considering the influence of different paraeters such as the edge conditions, teperature rise and follower axial forces (oy et al. 999, Pradhan et al., Haddadpour et al. 7, Torki et al. ). Vibration aplitudes of cylindrical shell ay not, however, sall enough to be accurately predicted by the linear analysis and thus the geoetric nonlinearity should be accounted for. This has been the ain concern of any studies in recent years. The work of Mahoudkhani et al. () is one of the first attept in this area which is devoted to the analysis of nonlinear priary resonance response of FG cylindrical shells. The ultiode Galerkin s procedure together with the ethod of ultiple scales (MMS) is used to obtain the frequency-response curves and identify the ranges of excitation aplitudes and frequencies in which the ulti-ode response occurs. The effects of teperature and axial loads on the free and forced nonlinear vibrations of FG cylindrical shells is investigated by Sheng and Wang () and Bich and Xuan Nguyen (), using again the Galerkin s ethod. Rafiee et al. () investigated the piezoelectric FG cylindrical shell under aerodynaic and theral loads. Sofiyev (6) studied the vibration of orthotropic FG cylindrical shells, giving an expression by the Jacobian elliptic function for the nonlinear frequencies. The hootopy perturbation ethod (HPM) is used to solve the single tie equation obtained fro the Galerkin s ethod. Cylindrical shells with different siplysupported, claped and free boundary conditions are studied by Strozzi and Pellicano (). They used the Chebyshev orthogonal polynoial for discretization of the equations. Jafari et al. () studied the FG cylindrical shell ebedded with a piezoelectric layer and used the Multi-ode Galerkin and Runge Kutta ethod to provide the solution. More detailed inforation and references on this subject ay be found in the review paper by Alani and Aabili (). The ethod of solution adopted in ost studies as those entioned above, begins with discretizing the governing partial differential equations (PDEs) with the odal expansion ethod and then uses either the analytical or nuerical ethods to deterine the periodic response. Aong the analytical ethods, the MMS is ost frequently used due to its established routine. ower order MMS ay not, however, accurate enough as the oscillation aplitude increases. Using higher order MMS ay also require heavy atheatical anipulation. Soe other analytical ethods based on the hoogony ethod such as HPM or Hootopy analysis ethod (HAM) can give ore accurate first order approxiation (He 999, iao ) and their extension to higher orders can be readily perfored. Hence they are proper choices for accurate and efficient solution of tie equations especially when applied to a single equation. The single tie equation obtained by using a single linear ode in the Galerkin s ethod is not, however, reliable and ay yield erroneous result. Alternatively, the nonlinear noral odes (NNM) can be eployed to decouple the set of nonlinear equations obtained by the ulti-odal Galerkin s ethod. The single nonlinear equation that will be obtained in this way, could correctly describe the nonlinear behavior of the syste. The concept of NNM is first introduced by Rosenberg (96) as a periodic synchronous oscillation of all degrees of freedo (DOF) of a nonlinear syste. Since then, extensive studies have been atin Aerican Journal of Solids and Structures (7) -

3 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells perfored both on their applications and also their ethods of coputation (Vakakis 997, Pierre et al. 6, Kerschen et al. 9, Peeters et al. 9). The invariant anifold (IM) approach, proposed by Shaw and Pierre (99) is one of these ethods that defines the NNM as a twodiensional invariant anifold in phase space. The ethod extends the invariance property of the linear noral odes to nonlinear systes and ay be considered as the generalization of the Rosenberg s definition (Peeters ). It is to be noted that in contrast to the linear noral odes, the NNMs are not orthogonal, but due to their invariance property they could still be suitable for accurate and efficient order reduction in nonlinear oscillatory systes (Kerschen et al. 9). Moreover the ode shapes corresponding to NNMs would change with aplitude and thus with tie (Touze et al. ). IM ethod is applied in the present study to copute the NNMs of the FG cylindrical shell. For this purpose, the equations of otion, which are based on the Donnell s shell theory, are first discretized using the ulti-ode Galerkin s ethod. The obtained ordinary differential equations are then decoupled using the IM ethod and then solved by HAM to deterine the frequencyaplitude relation. Nuerical results are obtained for backbone curves and nonlinear invariant surfaces. Nonlinear ode shapes are also presented by depicting the out-of-plane displaceent variations in axial and circuferential directions. FORMUATION According to the siple power low, the variation of volue fractions of the ceraic and etal through the thickness direction ( z ) can be described as, V N æ z h / ö æ /, Vc z h ö -, ç è h ø è ç h ø N () where V and V c represent the volue fractions of etal and ceraic, N is the volue fraction index and h is the thickness of the shell. The effective aterial properties of FG shell ( P eff ) including Young s odulus, E, density, r and Poisson s ratio, n, are defined as, Peff P V PV c c, () where P and P c denote the aterial properties of etal and ceraic. Using Eqs. () and () the effective aterial properties can be expressed as a function of z, P eff - z h / N Peff Pc ( P - Pc )( ). () h It is evident fro Eq. () that for N, the volue fraction of ceraic becoe zero and so P i.e. the aterial coposition is pure etal and with the increase in N, the aterial varies fro pure etal to pure ceraic. atin Aerican Journal of Solids and Structures (7) -

4 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells 5. Governing Equations Considering the coordinate syste shown in Figure, the equation of otion and the copatibility equation of the FG cylindrical shell can be expressed in ters of redial displaceent,w and the airy stress function, F as (Jansen 8), F ( w) - ( F) - - (, ), N w F R s w ( F) ( w) (, ), N w w R s () (5) Where * * * * 66 () D ( D D ) D, s R s q R q * * * * * - 66 () B ( B B B ) B, s R s q R q * * * * 66 () A ( A A ) A, s R s q R q () (),, N (, ) T S T S T S T S - s R q R s q R s q R q s (6) and, A * - * - A, B BA, D D- BA B (7) * - where A, B and D are the sub-atrices of the stiffness atrix and are defined as, h / -h / ( A, B, D ) ò Q (, z, z ) dz, (8) where Q are the coponents of the stiffness atrix based on the plane-stress Hook s law whose nonzero coponents are defined as, Ez () n() zez () Ez () Q Q, Q, Q 66. (9) -n( z) - n( z) ( n( z)) It ust be noted that to drive the copatibility equation, Eq.(5), the in-plane inertial ters should be neglected and the stress function, F, would be defined in the way to satisfy the in-plane equations of otion as, F F F q sq - R s Ns, N, N R q s q. () atin Aerican Journal of Solids and Structures (7) -

5 6 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells z,w s,u h z,w,v R Figure : FG cylindrical shell geoetry. In which N s and N q are the noral stress resultant in axial,s, and circuferential directions, q, respectively. Also, Ns q is the shear stress resultant.. Discretization of PDEs The nonlinear PDEs obtained for the equations of otion (i.e., Eqs. () and (5)) are first discretized using the Galerkin s ethod with appropriate trial functions. Various approxiations are used for this purpose in the literatures, aong the the function used by (Aabili et al. 998) contains the iniu nuber of the linear odal functions needed for accurately predicting the softening behavior of the shell. This function is defined as, ps ps w W( t)cos( nq)sin( ) W( t)sin( ) ps ps W()sin( t ) cos( nq) W()sin( t ), () where, W () t s are the unknown generalized coordinates to be deterined. Substituting the displaceent function, Eq. (), into the copatibility equation, Eq. (5), yields a linear nonhoogeneous differential equation in F that its particular solution, F, can be obtained as, p ps F f ( t) éf ( t)cos( n q) f ( t)cos( n q) f ( t)sin( n q) ùsin( ) p êë úû ps é f ( t)cos( n q) f ( t)cos( n q) f ( t)cos( n q) ù êë cos( ) 5 6 úû ps i ps f ( t)cos( n )sin( ) f cos(i n ) f cos( ) q 7 å q i 7 å i i i (i- ) ps å f sin( ) i i () where fi () t s are functions of W () t s. The hoogenous solution, F, can also be written as, h atin Aerican Journal of Solids and Structures (7) -

6 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells 7 where N s, N q and F h s s, N R q N q s - N q R q s () Ns q are stress resultants that arise fro the constraints on the edges of the shell and can be deterined in ters of Wt s i() by applying the average in-plane boundary conditions and the continuity requireent of circuferential displaceent, v (Aabili et al. 998). The above equation is not a general hoogeneous solution and is chosen to satisfy the average in-plane boundary conditions which are defined for the classic siply supported cylindrical shell (i.e., v and N ) as follows (Aabili et al. 998), s, F N Rdsd or Rdsdq, R q p p s q ò ò ò ò () F N q Rdsdq or - Rdsdq. R s q p p s ò ò ò ò (5) The continuity of v is also fulfilled via the following relation, p v Rdsd q, q ò ò (6) where v q can be related to w and F as, * * * * v F F w w w A A R B R B w - q R q s s R q R q ( ). (7) In the above equation F ust be replaced by the coplete solution of copatibility equation, i.e., F F F where F and F are deterined by Eqs. () and (). Then using Eq. () p h p h N q and Ns q can be obtained in ters of the generalized coordinates as, nw t W t s q, q 8RA * RA * N s, () () N N N - (8) s Finally, introducing Eqs. (), () and () into Eq. (), and applying the Galerkin s ethod, four nonlinear ODEs will be obtained as, dw() t w W () t å CqsW () tw q() tw s() t dt qs,,, QqW () tw q() t, (9) å q,, (, ) (,),(, ),(,),(,), atin Aerican Journal of Solids and Structures (7) -

7 8 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells where w s are linear natural frequencies corresponding to the odes used in Eq. () and C qs and Q are the constants that depend on the aterial and geoetric properties of the cylindrical q shell.. Extraction of Nonlinear Noral Modes Eqs. (9) include coupled quadratic and cubic nonlinear ters that ay be decupled via the corresponding NNMs. To extract the NNMs, the IM approach is used by expressing all the generalized coordinates, Wt s () and their tie derivatives, Wt () s, in ters of W and W. This pair, which i i is referred to as the aster coordinates, corresponds to the fundaental ode of the shell. Using the definition, u W (), t v W () t, the following fifth order polynoial expansion is used to approxiate the slave coordinates: (-k) k (-k) k (-k) k () å ( k ) å ( k ) å ( k 6) k k k 5 (-k) k (5-k) k åa( k ) u v åa( k 5) u v, k k (-k) k (-k) k (-k) k () å ( k ) å ( k ) å ( k 6) k k k 5 (-k) k (5-k) k åb( k ) u v åb( k 5) u v, k k W t a u v a u v a u v Y t b u v b u v b u v (, ) (,),(, ),(,),(,), () where Y () t W () t, and a and b k k are the unknown constants that could be deterined by substitution of Eq. () into the state-space for of the Eqs. (9) and equating the coefficients of like powers of u and v. The final fifth order expressions that will be obtained for W () t are lengthy and ay not be given here. But to give an insight to the characteristics of the NNMs, the third order approxiation is presented as follows: ( w - w ) S S ( w -w ) ( w - w ) w - w w -w W W W P( ) W P( ) W () tw p p (,,) (9 w w )( w w ) ( w w ) (, ) (,),(, ),(,)., () where S, P and P are constant ters that depend on the syste paraeter values. It is evident fro Eq. () that the fundaental NNM of the cylindrical shells ay depend on both aplitude and velocity. This behavior, which is the coon characteristic of nonlinear systes with quadratic nonlinearities, iplies that every point of the structure in a certain NNM does not reach the zero-deforation state at the sae instant (Nayfeh and Nayfeh 995). Substituting Eq. () atin Aerican Journal of Solids and Structures (7) -

8 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells 9 into Eq. (9), the uncoupled odal ODEs associated with the fundaental NNM of the cylindrical shell can be obtained as follows: where, W () t W () t GW (, W ), () w GW (, W ) kw ( t) kw ( t) kw ( tw ) ( t) kw 5 kw 5 () tw () t kw 6 () t kw 7 kw 8 () tw () t kw 9 () tw () t, () with k i s being constant ters dependent on the aterial and geoetric properties of the shell.. Solution by HAM An efficient and accurate solution can be provided for Eq. () using the HAM (iao ). For this purpose, the transforation, t w t and X() t W() t - d with w and d being the nonlinear N frequency and the constant drift aplitude respectively, are used as follows, w X ( t) w X ( t) G [ X ( t) d, w X ( t)]. () The unknown drift is included in the response, since Eq. () has quadratic and quartic ters. Considering the initial condition of X() W and X (), it is reasonable to choose the following function as an initial guess of the solution, The nonlinear operator, Ν is then defined as, N X () t W cos(). t (5) ft q ( ; ) ft ( ; q) Ν [ ft ( ; q), W( q), D ( q)] W ( q) w ft ( ; q) G[ ft ( ; q) D( q), W( q) ], (6) t t where q is the ebedding paraeter that varies fro to and W () q, D () q and ft (;) q are the functions that represent the solution of the ODE when q. The following linear operator is also defined in such a way that it s hoogeneous solution is cos( t ), ft (;) q [ ft ( ; q)] w [ ft ( ; q)]. (7) t with w being the first order approxiation of w N. The zeroth-order deforation equation can now be defined as, ( -q) [ ft ( ; q) - X ( t)] qkn [ ft ( ; q), W( q), D( q) ] (8) atin Aerican Journal of Solids and Structures (7) -

9 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells where k is the auxiliary undeterined constant paraeter that if chosen properly ay iprove the convergence of the solution. Next, W () q, D () q and f (; tq) are expanded in the Taylor series as, f(; tq) å å! q q W() q å w å w! q q D() q å d å d! q q f(; tq) f(;) t q X( t) X ( t) q W () q W () q q D () q D () q q (9) which converges to the exact solutions of w, d and X() t when q. That is to say, N f(;) t X( t) X ( t) X ( t), W () w w w, N å å D () d d d å () where X () t, d and w are the unknowns that could be defined via constructing the higher-order deforation equations by differentiating the zeroth-order deforation equation ties with respect to q, then dividing it by! and finally setting q. The resulted th order deforation equations is, [X ( t) - c X ( t)] kh( t) R [ X ( t)] () ( -) - where c for and c for > and, - N[ f( tq ; ) D( q), W( q)] R[ X( t)] ( - )! - q q () Eq. () is a linear differential equation that along with the initial conditions, X () and X () could be readily solved to obtain higher order approxiations of the solution. This is perfored along with eliinating the secular ters (i.e., coefficients of cos( t )), leading to algebraic equations in w and d. In nuerical analysis of the present proble, it is found that the values of d are negligible for various syste paraeters. This is because that the coefficients of quadratic and quartic ters in Eq. () are uch saller than those of cubic ters and can be neglected. Hence d s are set to zero, which leads to the following relation for w : atin Aerican Journal of Solids and Structures (7) -

10 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells w w a ( k7a ka w ) ( 5ka 5ka -) w 8 () where a W. The first order approxiation of the displaceent is also obtained as: X( t) 5ka é( k7 k8w) a ( kw k) ùcos( ) ê t w ë úû 6 5ka é êë úû w 5ka ( k8w - 5k7) a ( kw - k) ùcos( t) ( k8w -k7) cos(5 t) w 96 é 7 5 ù 5k9kwa - sin( t) sin( t) - sin( t) sin(6 t) ê ë 6 56 ú û () The first order solution presented by Eqs. () and (), is adequately accurate for the range of initial displaceent aplitudes considered in the present study. Arbitrarily higher order solutions can however be readily obtained using the entioned procedure, albeit their corresponding expressions are so lengthy to be given here. NUMERICA RESUTS AND DISCUSSION Nuerical results are presented in this section to exaine the accuracy of the IM ethod used for extraction of NNMs and also the solution of the obtained single nonlinear equation by the HAM. The nonlinear frequency-aplitude of FG cylindrical shells and also the nonlinear ode shapes of the cylindrical shell are also presented and copared with their linear counterparts. For the verification process it should considered that the solution provided in the present study contains three types of approxiation. The first approxiation corresponds to the discretization of the partial differential equations (i.e., Eqs. () and (5)) which have led to the discretized equations defined by Eq. (9). The second approxiation corresponds to the application of the IM approach to Eq. (9), and the third one is related to the analytical solution provided by using the HAM. Therefore, to assess the validity or accuracy of the results obtained by each of these approxiations, three nuerical studies are perfored in Figures, and 5. To ensure the validity and convergence of the discretization process used for obtaining Eq. (9), the nonlinearity index, a defined in æw ö N Mahoudkhani et al. () by a -, is calculated for a cylindrical shell ade up of a ç w è ø the stainless steel and the silicon nitride with n 5, / R, h / R., R., and different values of N. The results are copared with the result of Mahoudkhani et al. (), as shown in Figure. A close agreeent can be seen between the results which can ensure the validity of the present solution. atin Aerican Journal of Solids and Structures (7) -

11 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells Mahoudkhani et al. Present N Figure : Nonlinearity coefficient, a, v.s. the volue fraction index, copared with Mahoudkhani et al. (). Next the accuracy of the NNM extracted by the IM approach fro the discretized equations is exained in Figure, by coparing the fundaental nonlinear ode s response, obtained by the nuerical solution of Eq.(), with the results of nuerical siulation of the set of four equations given in Eq. (9). The syste paraeters used for siulations are n, / R, h / R., R.5. The shell is also assued to be coposed of stainless steel and silicon nitride with the aterial properties given in Table. For the initial conditions (ICs), W () a and W () are used and the reaining ICs, required for solution of Eqs. (9), are obtained using the IM approach to deterine W () ( (, ) ¹ (,) ) in ters of W and W (i.e., the fifth order version of Eq. ()). The results, given in Figure atch well for a / h lower than.5 even when the third order IM is used. For higher values of initial displaceent, it is seen that the fifth order IM reains accurate when a / h grows to. Material P P - P P P E.e9.79e- -6.5e-7 Stainless steel a.e e- n.6 -.e-.797e-7 r 866 E 8.e9 -.7e-.6e e- a 5.87e e- Silicon nitride n. r 7 Table : Material properties of FGM s fro Reddy and Chin (998). atin Aerican Journal of Solids and Structures (7) -

12 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells Solution of Eq. obtained by rd order IM) Solution of Eq. obtained by 5th order IM) Solution of set of Eq. (9) Solution of Eq. obtained by rd order IM) Solution of Eq. obtained by 5th order IM) Solution of set of Eq (9) W W - - The HAM, then is used to calculate the nonlinear backbone curves of the cylindrical shell with the sae geoetric properties of the previous study and different values of the volue fraction in t (sec) (a) t (sec) (b) Figure : Coparison of the nuerical solution of the single equation obtained by the third and fifth order IM (Eq. ()) by the nuerical solution of set of four equations, Eq. (9), for (a) W () / h.5 (b) W () / h. The accuracy of the HAM solution is also assessed in Figure 5 by coparing the nuerical solution of Eq. () with the HAM solution. The syste paraeters are the sae as those used in the previous study. To iprove the convergence of the solution, the variations of the estiated nonlinear frequency and also the response aplitude at an arbitrary tie instant with k are depicted first in Figure Different-order HAM approxiations are included in this figure. The best accuracy ay be obtained by choosing k fro the region where the variation of the solution with k is inial. Hence k - ight be proper choice for the analysis. Tie histories obtained by different order HAM for a / h is copared in Figure 5-(a) with the nuerical solution. Excellent agreeent is seen in Figure 5-(a) with even first order HAM. To ephasize on the accuracy of the HAM for higher initial displaceent, results obtained for a / h 6 are also given in Figure 5-(b), which shows that the fourth-order HAM have copletely atched with the nuerical result N /.95 W (t ).9 N / (th order HAM) W (t )(thorderham) N / (rd order HAM) W (t )(rdorderham) Figure : Variation of the frequency ratio and the response aplitude at tie t. sec with the auxiliary paraeter k. atin Aerican Journal of Solids and Structures (7) -

13 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells dex, N. The variation of the nonlinear frequencies ( w N ) with nondiensional aplitude ( a / h ) is given in Figure 6-(a). As N increases it is seen that the corresponding frequencies increase due to higher contribution of the ceraic part in the FGM. All the curves also show the softening behavior. To better illustrate the effect of N on the strength of the softening nonlinearity, w / w v.s. N a / h is depicted in Figure 6-(b). The slight increase in the bending of the curves is greater for interediate values of N ( N, 5 ). Siilar study is also perfored in Figure 7 for other syste paraeters taken as, n 8, / R, h / R., R. The nonlinearity is ostly hardening in this case, which is copletely different fro the previous one. The curves obtained in Figure 7- (b), also show the considerable effect of N on the nonlinearity. As is evident, for soe interediate values of N ( N 5, ), w / w initially decrease with aplitude and then, after a certain aplitude, begins to rise. For other values of N, the ost strong nonlinearity (i.e., bending of the N backbone curve) occurs for N which decreases as N grows. Nuerical solution st order HAM 6 Nuerical solution st order HAM th order HAM W W t (sec) -6.. t (sec) (a) (b) Figure 5: Displaceent tie history of the fundaental NNM aplitude obtained by the nuerical ethod and HAM for (a) W () / h (b) W () / h 6..5 N N N5 N N5 N.5 N N N5 N N5 N a a N (a) N / (b) Figure 6: (a) Aplitude v.s. fundaental NNM with w (b) Aplitude v.s. w / w. Corresponding to the N N n, / R, R.5, h / R.. atin Aerican Journal of Solids and Structures (7) -

14 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells a.5.5 N N N5 N N5 N a.5.5 N N N5 N N5 N N N / (a) (b) Figure 7: (a) Aplitude v.s. w (b) Aplitude v.s. w / w. Corresponding to the N N fundaental NNM with n 8, / R, R, h / R.. The invariant nonlinear odal surfaces corresponding to the fundaental NNM of the shell are next obtained in Figures 8 and 9, for the two cases of the cylindrical shells studied above. These surfaces depict the variation of the linear odal aplitudes, W, W and W with W and W. The results presented in Figure 8 for the first case study, show that W and W have a sall dependence on W although they considerably varies with W. It is in contrast with W, which has coparable dependency on both W and W. However, the contribution of the ode associated with W to the NNM is not considerable as the agnitude of W is uch saller than the agnitudes of W and W. This iplies that the axisyetric odes are ainly responsible for the softening behavior observed in Figure 6. The invariant surfaces of the second case study are shown in Figure 9. Coparing with the result of Figure 8, W is seen to have ore influence, in this case, on W and W, although the effect of W is still doinant. Moreover, W has the greatest aplitude and thus the corresponding ode is the ost doinant ode in the NNM copared to the axisyetric odes. This ode is, therefore, the ain cause of the hardening type nonlinearity seen in Figure 7. Also, due to the considerable effect of W on W, and also considering that W and W correspond to the odes with one longitudinal half wave nuber and different circuferential wave nubers, it ay be inferred that the nonlinear ode shape ay significantly vary with tie along the circuferential direction. This is exained in the next nuerical studies in Figures and. In these figures, the nonlinear ode shapes of the cylindrical shell is copared with their linear counterparts. atin Aerican Journal of Solids and Structures (7) -

15 6 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells. W.. -. W Y.5.5 W.5.5 Y.5.5 W (a) (b).5 W Y.5.5 W (c) Figure 8: The nonlinear invariant odal surface obtained for fundaental NNM by fifth order IM (a) W / h (b) W / h (c) W / h v.s. W / h and W / w h for N,, /, /.,.5. n N R h R R As is previously entioned, the nonlinear ode shapes ay not reain the sae at different instants. Hence the ode shapes in Figures and are presented for three different tie instants. Moreover at each tie instants, the displaceent distributions in longitudinal and circuferential directions are given in the sae two diensional plot. For the longitudinal distribution, q is set to zero and for the circuferential direction, s is set to /. The linear ode shapes are also included in these figures to illustrate the effects of the nonlinearity on the ode shapes. Results in Figure correspond to the first case study, whose corresponding invariant surfaces are presented in Figure 8. The ode shapes at t where W is zero, are shown in Figure -(a). Slight differences between the linear and nonlinear ode shapes, both for longitudinal and circuferential directions, can be recognized, although the overall shapes are the sae. Mode shapes at t.5t N where T p / w is depicted in Figure -(b). The displaceent aplitude ( W ) is nearly zero at this N N atin Aerican Journal of Solids and Structures (7) -

16 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells 7 point, while W is close to its axiu value. The nonlinear ode s displaceent distribution in both longitudinal and circuferential direction deviate considerably fro their linear counterparts such that the nuber of circuferential half-wave nubers are doubled at this instant. This happens due to the fact that the aplitude of W, which corresponds to the ode with n circuferential half-wave nuber, becoes doinant at this tie. However, since W have uch lower aplitudes in ost ties than the odes with n circuferential half-wave nubers (see Figure 8), thus the nonlinear ode shape seen in Figure -(b) can be observed only in a sall interval of tie. The ode shapes at t.5t N are also siilar to those at t, especially in the circuferential direction W.. 5 Y W 5 W Y 5 5 W (a) (b).5 W Y W (c) Figure 9: The nonlinear invariant odal surface obtained for fundaental NNM by fifth order IM W / h (b) W / h (c) W / h v.s. W / h and W / w h for N n 8 N / R, h / R., R,,, atin Aerican Journal of Solids and Structures (7) -

17 8 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells.5 -. Nonlinear ode inear ode -.5 Nonlinear ode inear ode w w w Nonlinear ode inear ode s/ s/ s/ (a) t (b) t.5t N (c) t.5t N Figure : Fundaental nonlinear ode shape of the cylindrical shell in different instants for n N, / R, h / R., R.5,, Mode shapes of the second case study, shown in Figure, exhibits a rather different nonlinear ode shapes in coparison to the first case. The displaceent distribution in the longitudinal direction is nearly the sae as the linear ode shapes, at different instants of tie. However, the distributions in the circuferential direction are very different fro the linear odes, such that at t and.5t N the ode shapes are siilar to the trapezoidal for rather than the sinusoidal for. This is in fact due to larger aplitudes of W as is seen in Figure 9. Moreover, the circuferential wave nubers at t.5t N (where the velocity has its axiu value), is twice the wave nuber of the linear ode shape. This is siilar to what is observed for the first case in Figure - (b), except that the ode s axiu aplitude is uch larger in this case and the nonlinear ode shape with n circuferential half-wave nuber is uch ore pronounced Nonlinear ode inear ode w.5 Nonlinear ode inear ode w.5 Nonlinear ode inear ode w s/ s/ s/ (a) at t (b) at t.5t N (c)at t.5t N Figure : Fundaental nonlinear ode shape of the cylindrical shell in different instants for n 8, N, / R, h / R., R. atin Aerican Journal of Solids and Structures (7) -

18 S. Mahoudkhani / Nonlinear Vibration and Mode Shapes of FG Cylindrical Shells 9 CONCUSIONS The nonlinear ode shapes of the FG cylindrical shell was calculated using the fifth-order IM approach and was eployed along with the HAM to provide an expression for the frequencyaplitude relation of FG cylindrical shells. For this purpose, the governing PDEs were discretized using the ost influential linear odes of the structure. These equations were then decoupled using the NNM, leading to a single nonlinear ordinary equation for the fundaental ode. The HAM was applied to solve this equation which was shown to give an accurate result even with the first approxiation, in the reasonable range of displaceent aplitudes considered in the present study. Nuerical studies were perfored to exaine the accuracy of the IM and HAM. Moreover the backbone curves, the nonlinear invariant surfaces and also the nonlinear ode shapes of the FG cylindrical shell were obtained for two different thickness to radius ratios. The softening nonlinearity with inial dependence on N was observed for the thicker shell. The thinner cylinder, however, exhibited the hardening behavior with the ost nonlinearity occurred for lower values of N. It was found that the axisyetric ode with twice circuferential wave nuber (i.e., ode (,)) is the doinant ode, copared to the axisyetric odes, that contributes to the NNM in the thinner cylinder. Moreover, the dependence of (,) ode s aplitude on both the displaceent and velocity of the ode (,) was found to be coparable, in contrast to the axisyetric odes that were ostly dependent on the displaceent. Hence in ties that the displaceent and velocity of the (,) ode reach their zero and axiu values, respectively, the aplitude of the ode (,) becoes doinant leading to the nonlinear ode shapes with circuferential wave nuber twice those observed for the linear ode shapes. This was in fact ore pronounced for the thinner cylinder. References Alani, F., Aabili, M., (). Non-linear vibrations of shells: A literature review fro to. International Journal of Non-inear Mechanics 58:-57. Aabili, M., Pellicano, F., PaÏDoussis, M.P., (998). Nonlinear vibrations of siply supported, circular cylindrical shells, coupled to quiescent fluid. Journal of Fluids and Structures (7): Bich, D.H., Xuan Nguyen, N., (). Nonlinear vibration of functionally graded circular cylindrical shells based on iproved Donnell equations. Journal of Sound and Vibration (5): Haddadpour, H., Mahoudkhani, S., Navazi, H.M., (7). Free vibration analysis of functionally graded cylindrical shells including theral effects. Thin-Walled Structures 5(6): He, J.-H., (999). Hootopy perturbation technique. Coputer Methods in Applied Mechanics and Engineering 78( ): Jafari, A.A., Khalili, S.M.R., Tavakolian, M., (). Nonlinear vibration of functionally graded cylindrical shells ebedded with a piezoelectric layer. Thin-Walled Structures 79:8-5. Jansen, E.., (8). A perturbation ethod for nonlinear vibrations of iperfect structures: Application to cylindrical shell vibrations. International Journal of Solids and Structures 5( ):-5. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F., (9). Nonlinear noral odes, Part I: A useful fraework for the structural dynaicist. Mechanical Systes and Signal Processing ():7-9. iao, S. (). Beyond Perturbation: Introduction to the Hootopy Analysis Method, CRC Press. atin Aerican Journal of Solids and Structures (7) -

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