Linear and forced nonlinear vibrations of shallow shells with two simply supported and two free edges

I.D. Breslavsky, McGill University, Department of Mechanical Engineering, Montreal, Canada

K.V. Avramov, A.N. Podgorny Institute for Mechanical Engineering Problems, Department of Nonstationary Vibrations, Kharkov, Ukraine

 

 

Summary. The exact eigenmodes of shallow shells with simply supported opposite edges is obtained. Eigenmodes of linear vibrations of shallow shell with two simply supported and two free edges are treated. The vibrations of shallow shells with geometrical nonlinearity in the case of an internal resonance are described by two-degrees-of-freedom system with essential nonlinearity. The interaction of two shell modes is described by this system. The harmonic balance method is used to study this system. The stability of shell periodic motions is analyzed. The shallow shell bifurcation behaviour is analyzed; symmetry-breaking and saddle- node bifurcations are observed.

 

 

 


1 Introduction

 

Shallow shells are elements of turbines, aircrafts and marine structures. Significant dynamical loads frequently acts on these structures, which can lead to their failure. Shallow shells perform lateral vibrations with amplitudes, which are commensurable with the shell thickness. Then the shells motions are described using geometrically nonlinear theory. Many efforts were made to study vibrations of shallow shells with geometrical nonlinearity. Grigolyuk [1] investigated the vibrations with moderate amplitudes of simply supported panel using two-mode expansion. Cummings [2] considered the vibrations of simply supported cylindrical panel under the action of impacts loads. Using single-mode approximation, Leissa and Kadi [3] investigated nonlinear free vibrations of simply supported shallow panel with different values of the principal curvatures in two orthogonal directions. The nonlinear vibrations of simply supported cylindrical panel are analyzed by Volmir, Logvinskaya, Rogalevich [4]. Three and five degrees-of-freedom models are treated. Free vibrations of clamped cylindrical panels are analyzed in the paper [5]. The spline finite strip method is developed to analyze free vibrations of doubly curved shallow shells with rectangular base in [6, 7]. Liew and Lim [8] analyzed linear free vibrations of rectangular base shallow shells with different curvatures and different boundary conditions. The middle thickness shallow shells are analyzed accounting shear by Liew, Lim [9]. Baumgarten and Kreuzer [10] analyzed two- and four-modes models of cylindrical panel nonlinear vibrations. Yamaguchi and Nagai [11] investigated chaotic vibrations of square base shallow shells with flexible edges using finite degrees-of-freedom shell model. Awrejcewicz and Krysko [12] analyze the period-doubling bifurcations and chaotic motions of rectangular plate. The forced vibrations with moderate amplitudes of doubly curved panels with rectangular base in the case of the internal resonance are considered in [13]. Nonlinear vibrations of shallow shells with complex boundaries are treated in [14, 15]. The influence of the initial imperfections on the nonlinear vibrations of shallow shells is analyzed by Amabili [16-18]. The exact eigenmodes of the rectangular plate with two simply supported opposite edges are obtained in [19]. The similar expressions for open cylindrical shells are derived in [20]. Free vibrations of Levy- type thick functionally graded cylindrical shell panels are investigated to identify the validity range of Donnell and Sanders theories in [21]. Free vibrations of two-dimensional functionally graded open cylindrical shell are analyzed using 2-D generalized differential quadrature method in [22]. Nonlinear modes of simply supported cylindrical shell are treated in [23].

Free and forced vibrations of shallow cylindrical panel, which is simply supported at the opposite edges and free at others edges, are analyzed. The exact eigenmodes of shallow shells with simply supported opposite edges are obtained. The effect of the shell parameters on the eigenfrequencies and eigenmodes is analyzed. It is shown that the first eigenfrequency does not depend on both the shell curvature and the length along simply supported edges. If the length along the simply supported edges is increased, the higher eigenfrequencies are decreased. The shell nonlinear vibrations with moderate amplitudes in the case of internal resonance are considered. Due to this internal resonance two eigenmodes of shell vibrations are interacted. In order to study this interaction essentially nonlinear two-degree-of-freedom system is obtained and analyzed by using the harmonic balance method. As a result of stability analysis, saddle-node and a symmetry-breaking bifurcation are analyzed.

 

2 Problem formulation

 

The forced nonlinear vibrations of simply supported shallow shell with rectangular base (Fig. 1) under the action of the concentrated force are considered. The double curved shallow shell is analyzed. As thin shell is treated, shear and rotation inertia are not taken into account. It is assumed, that the shell displacements are moderate; stresses and strains are small. Then stresses and strains satisfy the Hookes law; strains and displacements are connected by nonlinear relations. Then the Donnell equations with respect to the displacements are used [24, 25]:

(1)

(2)

(3)

where are displacements of the shell middle surface in directions, respectively; , are Youngs modulus, Poissons ratio and material density; and are curvatures of shell middle surface in x and y directions, respectively; is the shell thickness; are intensities of the external forces.

The edges ; are simply supported and the edges ; are free. Then the following boundary conditions are satisfied:

; ; ; ; (4)

; ; ; , (5)

where , , are membrane forces per unit length; , , are moments per unit length; are transverse shear forces [24]:

; ;

;

.

 

3 Free linear vibrations of shell

 

Nonlinear vibrations are expanded into the truncated series by eigenmodes. Therefore, before analysis of nonlinear problem, the shell eigenmodes are calculated. Then the nonlinear terms are not taken into account in the equations (1-3) and it is assumed that . In future analysis it is considered the doubly curved shallow shells with arbitrary boundary conditions at the edges ; . The boundary conditions (4) satisfy at the edges and .

Using the separation of variables and satisfying the boundary condition (4), the shell vibrations are presented as:

;

; (6)

,

where is positive integer.

The shell displacements (6) are substituted into the linearized Donnell equations (1-3). As a result, the following system of partial differential equations with respect to is derived:

(7)

(8)

(9)

The displacements are presented in the following form:

; ; . (10)

The solution (10) is substituted into (7-9). As a result, the system of ordinary differential equations with respect to is obtained. The solution of this system takes the following form:

; ; , (11)

where are constants of integration. The displacements

;

are substituted into (7-9) to obtain the parameters ,. As a result, the system of linear algebraic equations with respect to is derived:

(12)

The elements of the matrix are the following:

;

; ;

; ; ;

; ; .

The determinant of the system (12) is equated to zero and the following characteristic equation with respect to is derived:

(13)

One element of the eigenvector is taken equal to 1: . We stress, that the values and are complex.

The solutions (6, 10) are substituted into the boundary conditions (5). After the variables separation the following boundary conditions are derived:

;

;

;

.

(14)

If the solutions (11) are substituted into the equations (14), the homogeneous system of eight linear algebraic equations with respect to is obtained.

The solution of the equation (13) corresponding to the first eigenfrequency are the following:

(15)

where are numbers, which are obtained from the numerical solution of equation (13). The solutions of the equation (13) for higher eigenfrequencies take the form:

.

For the first eigenfrequency the eigenmodes of the lateral displacements takes the form (A.1) from Appendix. The eigenmodes and takes the form (A.2) from Appendix.

The solutions (A.1) and (A.2) are substituted into the boundary conditions (14). As a result the system of linear algebraic equations with respect to is obtained. The frequency equation with respect to is obtained by equating to zero the determinant of this system. The parameters are obtained from the system of linear algebraic equations. Using the above-considered approach, the exact analytical form of eigenmodes is derived. These eigenmodes are presented in Appendix.

The suggested approach differs from the method [24], where the last boundary condition (4) is not taken into account. We stress that the eigenmodes (6, 10, 11) satisfy all boundary conditions at the edges and .

 

4 Numerical analysis of linear vibrations

 

The steel shallow shell with the following parameters:  N/m2,  kg/m3, , a1=1 m, a2=0.6 m, h=0,01 m,  m is analyzed. The edges and satisfy the boundary conditions (5). Table 1 shows the results of eigenfrequencies calculations. The eigenfrequencies of the plates, which are obtained using the relations from [20, 26], are presented in the first row. Table 2 shows the dependence of the eigenfrequencies on the shell length at a1=1 m, m, h=0.01 m. For all considered parameters of the shell the eigenmode of the first eigenfrequency () have no nodal lines parallel to axis. The eigenmodes of the second and the third eigenfrequencies have one and two nodal lines parallel to axis, respectively. As follows from Tables 1, 2, the frequencies of the eigenmodes without nodal lines parallel to axis do not depend on the shell geometrical parameters; the frequencies of the eigenmodes with one nodal line parallel to axis do not depend on shell middle surface curvature.

The eigenfrequencies of the shallow shell are determined by the Rayleigh-Ritz method to verify the obtained results. The trial functions are used for the expansions of , which satisfying the boundary conditions (4). Table 3 shows the results of the eigenfrequency calculations for the shells with the parameters a1=1 m, a2=0.7 m, m, h=0,01 m. As can be seen from Tables 2 and 3, the eigenfrequencies, which are obtained by the Rayleigh-Ritz method, are close to the eigenfrequencies, which are obtained from analytical analysis.

As follows from Tables 1 and 2, internal resonances can occur in the shallow shells.

Fig. 2 shows three eigenmodes of the shell linear vibrations with the following parameters: a1=1 m, a2=0.7 m, m, h=0.01 m. These eigenmodes correspond to the eigenfrequencies: 154.9; 324; 619.9 rad/s.

 

5 Nonlinear vibrations analysis

 

The forced vibrations of the shell (Fig. 1) under the action of the point force are considered. This point force has the following projections:

,

where is delta-function.

The shell displacements fields are expanded by using the eigenmodes of linear vibrations [23, 14, 15]:

(16)

,

where are generalized coordinates of the shell; are normalized vibrations eigenmodes, which are determined in the previous section.

As thin shells are considered, the eigenfrequencies of longitudinal vibrations are significantly higher then the eigenfrequencies of bending vibrations. Therefore, the displacements are significantly less than [23, 14, 15] and the inertial terms are not taken into account in the equations (1, 2). The expansions (14) are substituted into the equations (1-3) and the Galerkin method is used. Then the equation (3) is reduced to the system of ordinary differential equations. The system of 2N linear algebraic equations with respect to is derived from the equations (1, 2). As a result the following equations are obtained:

; (17)

; (18)

, (19)

where the parameters are obtained from the numerical calculations. Solving the system of linear algebraic equations (20, 21), the generalized coordinates are obtained as functions of . These solutions are substituted into the system of ordinary differential equations (19) and the -degrees-of-freedom model of the shell vibrations is obtained.

The following dimensionless generalized coordinates and dimensionless time

; ;

are used.

It is assumed in future analysis that the frequency of the external force is close to the first eigenfrequency

, (20)

where is detuning parameter. As follows from Tables 1, 2, the eigenfrequencies satisfy to the condition of internal resonance:

, (21)

where is small parameter, is detuning parameter.

It is assumed, that the third and higher eigenfrequencies of the shell are significantly greater then . The frequency of the external force satisfies the equation (20). The eigenfrequencies and meet the resonance conditions (21). Then the third and higher vibrations eigenmodes can not be accounted in the expansion for (16). The shell vibrations are described by two-degree-of-freedom nonlinear dynamical system with respect to the modal coordinates :

(22)

The harmonic balance method is used to study the vibrations of this system. The shell vibrations are presented by the following truncated Fourier series:

. (23)

The solution (23) is substituted into the dynamical system (22) and the coefficients at the same harmonics are equated. Then the amplitudes satisfy the system of nonlinear algebraic equations. This system can be presented in the following general form:

(24)

The results of forced vibrations analysis are shown on a frequency response. The continuation technique [27] is used to obtain a frequency response. Then one of the parameters from is preset with certain step. The rest parameters are obtained by the solution of the system of nonlinear algebraic equations (24).

The stability of the motions is analyzed. Then small perturbations are added to periodic motions. Then small perturbations are described by the system of four linear ordinary differential equations with respect to . This system can be presented in the following vector form:

(25)

where is matrix with time periodic elements. The fundamental matrix is calculated to estimate the periodic motions stability [27, 28]. The fundamental matrix is solution of the system (25). The identity matrix is used for initial conditions of the system (25):

The system of linear differential equations with periodic coefficients (25) is solved four times to calculate the fundamental matrix. This system is solved numerically by the Runge- Kutta method. The multipliers, which are eigenvalues of the fundamental matrix, are calculated. Stability of periodic solutions is estimated by using of the multipliers [28, 29].

 

6 Numerical analysis of nonlinear vibrations

 

The numerical analysis will be carried out for steel shell with the parameters:

a1=1 m, a2=0.7 m, , m, h=0.01 m, E=2.1∙1011 N/m2, ν=0.3.

The nonlinear dynamics of the system (22) is studied. Note, that the quadratic and cubic terms contribute essentially into the system (22). The harmonic balance method is used to analyze the dynamics of the system (22).

Free vibrations of the system (22) are analyzed; thus . The vibrations modes with active coordinate and are described by the following dynamical system:

. (26)

The harmonic balance method is used to study the dynamics of the system (26); the system motions are presented as

(27)

The harmonic balance method is applied to analyze the forced vibrations in Section 5. Then the solution (23) contains the frequency of the external force . In this section the harmonic balance method is used to analyze free nonlinear vibrations. In this case the solution (27) contains unknown frequency .

The calculations with sequential increase of the number of terms in the expansion (27) are carried out to determine the sufficient number of terms to approximate nonlinear vibrations. It is determined that four terms of the expansion (27) are enough to approximate the vibrations. Fig. 3 shows by dot-and-dash line the backbone curve for the vibrations amplitudes .

Free nonlinear vibrations with two active coordinates and are observed in the system (22). Following the harmonic balance method, these motions are presented as:

, (28)

where is unknown frequency, which is determined. The harmonic balance method considered in the previous section is used. As a result of these calculations, the motions, which are denoted by number 2 on Figures 4, 5, are obtained. These vibrations have two active generalized coordinates and . In this regime the harmonics with the amplitudes and dominate in the solutions and , respectively.

Now the forced motions of the system (22) are analyzed. It is assumed, that the external force with amplitude  N is applied at the point . Then the parameters of the system (22) are: ; . The parameters is equal to zero as the second eigenmode satisfy the equation . The harmonic balance method is applied to the system (22).

The vibrations with and take place in the system (22). These motions are described by the following equation:

. (29)

Performing numerical solution of the system of nonlinear algebraic equations (24), two kinds of forced periodic vibrations are obtained. The curves of these motions are denoted by the numbers 3 and 4 on Fig. 3. Stable and unstable forced vibrations are shown by solid and dotted curves on Figures 3-5. Curve 4 on Fig.3 shows the amplitude of the first harmonics . The harmonics with the amplitude predominates in the vibrations, which are described by the curve 3 on the left from the point A. The motions, which are described by the branch 3 (Fig. 3) close to the point B, satisfy the symmetry condition [30]:

, (30)

where is vibration period. The vibrations, which are described by the curve 5 (Fig. 4, 5), are forked from the motions (30) due to the symmetry-breaking bifurcation [30]. This bifurcation behavior is determined on the basis of the numerical calculations of the multipliers.

Now the vibrations with two active coordinates are treated. These motions are described by the curves 5, 6 on Figures 4, 5. The harmonics with the amplitudes and predominate in the Fourier-series expansions of the generalized coordinates and , respectively. Note, that the harmonics with the amplitude contributes into the Fourier-series expansion of .

The direct numerical integration of the system (22) is carried out to verify the results, which are obtained by the harmonic balance method. The initial conditions for the direct numerical integration are chosen from the results, which are obtained by the harmonic balance method. The results of the direct numerical integration are shown by dots in Figures 3-5. Thus, the results of the harmonic balance method and the data of the direct numerical integration are close.

 

7 Conclusions

 

The exact analytical expressions for the vibrations eigenmodes of shallow shells with rectangular base and two simply supported edges are derived in this paper. The connections between the eigenfrequencies of the panels and the shell geometrical parameters are analyzed. It is shown that the first eigenfrequencies do not depend on both the shell curvature and the length along simply supported edges. If the length along the simply supported edges is increased, the higher eigenfrequencies are decreased. We come to the conclusion that the internal resonances can occur in such type of shallow shells.

Analysis of free and forced nonlinear vibrations of shallow cylindrical panel accounting internal resonances is performed. The backbone curves of shallow shells are hard. The stability analysis of the periodic motions is carried out. The saddle-node and symmetry-breaking bifurcations are observed in the shell.

 

Appendix

 

(A.1)

The eigenmodes and are denoted by and , respectively. They can be presented as:

(A.2)

Note, that the constants of integrations are real.

 

References

 

[1] Grigolyuk, E.I.: Vibrations of circular cylindrical panels subjected to finite deflection, Prikl. Mat. Mekhanika 19, 376382 (1955) (in Russian)

[2] Cummings, B.E.: Large-amplitude vibration and response of curved panels, AIAA J. 2, 709716 (1964).

[3] Leissa, A.W., Kadi, A.S.: Curvature effects on shallow shell vibrations, J. Sound Vib. 16, 173187 (1971).

[4] Volmir, A.S., Logvinskaya, A.A., Rogalevich, V.V.: Nonlinear natural vibrations of rectangular plates and cylindrical panels, Sov. Phys. Dokl. 17, 720721 (1973).

[5] Srinivasan, R.S., Bobby, W.: Free vibration of noncircular cylindrical shell panels, J. Sound Vib. 46, 4349 (1976).

[6] Cheung, Y.K., Li, W.Y., Tham, L.G.: Free vibration analysis of singly curved shell by spline finite strip method, J. Sound Vib. 128, 411422 (1980).

[7] Li, W.Y., Tham, L.G., Cheung, Y.K., Fan, S.C.: Free vibration analysis of doubly curved shells by spline finite strip method, J. Sound Vib. 140, 3953 (1990).

[8] Liew, K.M., Lim, C.W.: Vibration of doubly-curved shallow shells, Acta Mech. 114, 95119 (1996).

[9] Liew, K.M., Lim, C.W.: Vibration studies on moderately thick doubly-curved elliptic shallow shells, Acta Mech. 116, 8396 (1996).

[10] Baumgarten, R., Kreuzer, E.: Bifurcations and subharmonic resonances in multi-degree-of-freedom panel's models, Mecc. 31, 309322 (1996).

[11] Yamaguchi, T., Nagai, K.-I.: Chaotic vibrations of a cylindrical shell-panel with an in-plane elastic-support at boundary, Nonlinear Dyn. 13, 259277 (1997).

[12] Awrejcewicz, J., Krysko, V.A.: Feigenbaum scenario exhibited by thin plate dynamics, Nonlinear Dyn. 24, 373398 (2001).

[13] Amabili, M.: Non-linear vibrations of doubly curved shallow shells, Int. J. Non-Linear Mech. 40, 683710 (2005).

[14] Breslavsky I.D., Avramov, K.V.: Nonlinear modes of cylindrical panels with complex boundaries. R-function method, Mecc. 46, 817-832 (2011).

[15] Breslavsky, I.D., Strelnikova, E.A., Avramov, K.V.: Dynamics of shallow shells with geometrical nonlinearity interacting with fluid, Comput. Struct. 89, 496506 (2011).

[16] Amabili, M.: Nonlinear vibrations of circular cylindrical panels, J. Sound Vib. 281, 509535 (2005).

[17] Amabili, M.: Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections, J. Sound Vib. 298, 4372 (2006).

[18] Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates, New York: Cambridge University Press 2008.

[19] Leissa, A.W.: The free vibration of rectangular plates, J. Sound Vib. 31, 257293 (1973).

[20] Zhang, L., Xiang, Y.: Vibration of open circular cylindrical shells with intermediate ring supports, Int. J. Solids Struct. 43, 37053722 (2006).

[21] Hotisseini- Hashemi S., Ilkhani M.R.: Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel, Acta Mech. 223, 1101-1118 (2012).

[22] Aragh B.S., Hedayati H.: Static response and free vibration of two- dimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions. Acta Mech. 223, 309-330 (2012).

[23] Avramov, K. V.: Nonlinear modes of vibrations for simply supported cylindrical shell with geometrical nonlinearity, Acta Mech. 223, 279292 (2012).

[24] Bolotin V.V.: Vibrations in Mechanics, Vol. 1, Moscow: Mashinostroenie 1978. (in Russian)

[25] Volmir, A.S.: Nonlinear Dynamics of Plates and Shells, Moscow: Nauka 1972. (in Russian)

[26] Leissa, A.W.: Vibration of Plates, Washington: National aeronautics and space administration 1969.

[27] Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems, New York: Springer 1980.

[28] Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients, New York: Wiley 1975.

[29] Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, New York: Springer 1983.

[30] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, New York: Springer-Verlag 2004.

 

Author addresses:

Department of Mechanical Engineering, McGill University, Macdonald Engineering Building, 817 Sherbrooke St. West, Montreal, Quebec H3A 0C3, Canada (E-mail: id.breslavsky@gmail.com)

A.N. Podgorny Institute for Mechanical Engineering Problems, National Academy of Sciences of Ukraine, Kharkov, Ukraine, (E-mail: kvavr@kharkov.ua)

 


 

 

 

Fig. 1. Sketch of the shell

 

 


 

a. b.

c.

Fig. 2. Three eigenmodes of the shell linear vibrations with the parameters: a1=1 m, a2=0.7 m,  m, h=0.01 m; a., b., c. correspond the eigenfrequencies 154.9; 324 and 619.9 (rad/s), respectively

 

 


 

Fig. 3. Backbone curve and frequency responses for the first generalized coordinate ξ1.


 

Fig. 4. Backbone curve and frequency responses for the second generalized coordinate ξ2.


 

Fig. 5. Backbone curve and frequency responses for the second generalized coordinate ξ2

 


Table 1. The dependence of eigenfrequencies on curvature radius

, m

n=1

n=2

n=3

∞ [24, 28]

154.9

370.9

1223

619.8

915.7

1847

1394.7

1722.6

2750.5

20

154.9

369.8

1244.7

619.9

909.7

1848.8

1394.7

1708.4

2732.9

5

154.9

368.6

1701.6

619.9

909.4

2045.7

1394.7

1708.3

2795.7

2.5

154.9

366.6

2254.5

619.9

908.8

2330.6

1394.7

1708.2

2894.8

1.25

154.9

356.7

3981.4

619.9

905.3

3401.3

1394.7

1706.8

3340.8

 

 

 

 


 

Table 2. The dependence of eigenfrequencies on the shell length

,

n=1

n=2

n=3

0.3

154.9

683.8

2593.8

619.9

1476.2

3293.1

1394.7

2442.4

4041.3

0.5

154.9

428.1

2558.1

619.9

1011.9

2807.5

1394.7

1833.8

3451.7

0.6

154.9

366.6

2254.5

619.9

908.8

2330.6

1394.7

1708.2

2894.8

0.7

154.9

324

2066.6

619.9

840.6

1997.6

1394.7

1627.9

2530.4

1

154.9

251.9

1701.3

619.9

733.9

1393.2

1394.8

1508.1

1968.7

1.5

154.9

203.3

1234.9

619.9

670.5

964

1394.8

1441.3

1648

 


Table 3. The eigenfrequencies, which are obtained by Rayleigh-Ritz method, for the shell with the following parameters: a1=1 m, a2=0.7 m,  m, h=0.01 m

N=1

n=2

n=3

147.5

309.3

2064

598.9

766.1

1948.6

1394.7

1614

2588.9