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Mehdiyev Mahir Aligulu
Institute of Mathematics and Mechanics of NAS of Azerbaijan
Received: December 30, 2011 / Accepted: January 31, 2012 / Published: February 15, 2012.
Abstract: In the paper, a problem on parametric vibrations of a cylindrical shell contacting with external visco-elastic medium, stiffened by a longitudinally ribs and situated under the action of external pressure, is solved in a geometric nonlinear statement by means of the variation principle. Lateral shift of the shell is taken into account. Influences of environment have been taken into account by means of the Pasternak model. The curve separating the stability and instability domains of parametric vibrations has been constructed on the plane ?load-frequency?.
Key words: Ribbed shell, lateral shift, variation principle, nonlinear parametric vibrations.
1. Introduction??
Thin-shelled structural elements of envelope type compose a vast class of mechanical objects that are widely used in contemporary mechanical engineering, in space and rocket technology, and in construction. Strength analysis, stability and vibrations of such constrictions play an essential part in their designing. Nevertheless the behavior of thin-shelled constructions containing ribs that could take into account the discreteness of arrangement of ribs, shift and tensional rigidity of ribs, lateral shifts and geometric nonlinearity has not been studied enough. The reason for this is the complexity of the mentioned factors and necessity of solving bulky nonlinear boundary value problems. Furthermore, some constructions preserve carrying capacity after local loss of stability, and discovery of different forms of stability loss give rise to mathematical complexities. Therefore, development of mathematical models for investigating the behavior of stiffened shells that more completely take into account their work under dynamical loads, carrying out investigations of stability, and vibrations on their base, and also choice of rational parameters of a medium-contacting construction are urgent problems. The solution of such type problems represents some mathematical difficulty that becomes intensified both with regard to dynamical effects and influence of environment, where the elaboration of the approximate method is required. The variational method is one of them.
Ref. [1] has been devoted to the investigation of nonlinear deformation of cylindrical shells under the action of different type dynamical loads. Refs. [2-4] have been devoted to the investigation of stability, vibration and optimization of ridge cylindrical shells. The stability of cylindrical shells of step-variable thickness under the action of dynamical loads has been investigated by the variational-parametric method in Ref. [5].
The problems on parametric vibrations of a nonlinear and thickness inhomogeneous viscoelastic unstiffened filled cylindrical shell have been studied in Refs. [6, 7] by using the variational principle and the Pasternak model. Nonlinear vibrations of a stiffened, viscoelastic medium-contacting cylindrical shell have been researched in geometrically nonlinear statement by using the variational principle in Refs. [8-12].
In the present paper, by means of the variational principle, in geometrically nonlinear statement, for the first time we solve a problem on parametric vibrations of a longitudinally stiffened, viscoelastic medium-filled cylindrical shell subjected to the action of external pressure. The lateral shift of the shell is taken into account. The influence of environment is taken into account with the help of the Pasternak model. On the plane “load-frequency”, the curves separating the stability and instability areas of parametric vibrations are investigated. Influences of lateral shift on critical load parameter of shell’s stability are studied.
2. Problem Statement
We’ll obtain differential equations of motion and natural boundary conditions for a medium-contacting cylindrical shell longitudinally stiffened with regard to lateral shift on the base of Ostrogradsky-Hamilton variational principle. For applying the mentioned principle, we write beforehand the potential and kinetic energy of the system.
The potential energy of elastic deformation of the cylindrical shell with regard to lateral shift is of the form [13]:
(1)
The expressions for the potential energy of elastic deformation of the ?ith longitudinal rib are the followings [2]:
In Eqs. (1) and (2) x1, x2, y1, y2 are the coordinates of curvilinear and linear edges of the shell; Fi, Jzi, Jyi, Jkpi are the area and inertia moments of the cross section of the i-th longitudinal bar with respect to the axis Oz and the axis parallel to the axis Oy and passing through the gravity center of the cross section, and also its inertia moment under torsion; Ei, Gi are the elasticity and shift module of the material of the i-th longitudinal bar.
The potential energy of the shell, under the action of external surface and edge loads applied to the shell is determined as the operation performed by the loads when taking the system from the deformed state to the initial undeformed state, and is represented as follows:
(3)
The potential energy of external edge loads applied to the ends of the i-th longitudinal bar is similarly determined by the following expressions (it is assumed that only boundary loads are applied to the ribs):
??
(4)
The total potential energy of the system equals the sum of potential energies of elastic deformations of the shells and ribs, and also potential energies of all external loads:
A (5)
The kinetic energy of the shell and ribs is written in the form
? are densities of materials from which the shell and the i-th longitudinal bar has been made.
The kinetic energy of the ridge shell is defined as follows:
?
(8)
The equations of motion of a medium-contacting stiffened ridge shell are obtained on the base of Ostrogradsky-Hamilton principle on stationarity of the action: 0
W is the Hamilton action, ???KL is Lagrange’s function, t’ and t” are the given arbitrary times.
We can write the load intensity acting on the shell as viewed from the visco-elastic filler, in the following kc
(Pasternak model), where ▽2 is two-dimensional Laplace’s operator on the contact surface, w is the shell’s deflection, q0, q1 are constants.
Taking the equalities into account the constancy of radial deflections along the height of cross sections and the equality of corresponding twisting angels following from the conditions of rigid joint of ribs with the shell, we write the following relations:
??????;,,1ii
?, are the turning and twisting angles of cross-sections of longitudinal bars.
Allowing for Eq. (11), we express the displacement of bars by the displacement of the shell. From the stationarity Eq. (9) we get nonlinear algebraic equations with respect to desired unknowns.
3. Problem Solution
On an example consider nonlinear parametric vibrations of a longitudinally stiffened annular cylindrical shell with regard to lateral shift and under the action of radial load ,sin110tqqq??? where
? is the pressure change frequency of visco-elastic medium-filled shell. Assuming that the edges of the shell is hingely supported for lx; 0?
;
Where m is the number of waves in the peripheral direction, ??is the frequency of vibration of desired quantities:.
??tt integrate with respect to x, y and t. Then instead of Eq. (5) we get a function from the desired quantities,
4. Conclusions
The dependences of dynamical stability on the parameters of the construction on the plane“load-frequency” represented in the form of a curve are given in Fig. 1. This curve divides the plane into two domains: for the points of one domain the vibrations are restricted, for another one they are unbounded in time. In the graph, the vibrations of longitudinally stiffened cylindrical shell in visco-elastic medium are given by prime lines, by solid lines in elastic medium. Furthermore curve 1 corresponds to discount of lateral shift, in the shell, curve 2 to ignoring lateral shift in the shell. It follows from the calculation results that for a visco-elastic body, the breaking point of the typical curve rises over the frequency axis. Discount of influence of medium reduces to increase of stable zones of the shell, discount of lateral shift in the shell reduces to contraction of stable zones of the shell.
References
[1] V.D. Kubenko, P.S. Kovalchuk, N.P. Podchasov, Nonlinear vibrations of cylindrical shells, Vyshcha shkola, Kiev, 1989, p. 208. (in Russian)
[2] I.Y. Amiro, V.A. Zarutsky, Theory of ribbed shells. Methods of calculating shells, Naukova Dumka, Kiev, 1980, p. 367. (Russian)
[3] I.Y. Amiro, V.A. Zarutsky, V.N. Revutsky, Vibrations of RIbbed shells of Revolution, Naukova Dumka, Kiev, 1988, p. 171.(Russian)
[4] V.A. Zarutsky, Y.M. Pochtman, V.V. Skalozub, Optimization of Reinforced of Cylindrical Shells, Vyshcha Shkola, Kiev, 1990, p. 138. (Russian)
[5] D.I. Aristov, V.V. Karpov, A.Y. Salnikov, Variation-parametric method of investigation of cylindrical shells stepwise variable thickness under dynamic loadingiffened, Mathematical modeling, numerical methods and programs Intercollege Thematic, Sat trudy SPbGASU-SPb, 2004, pp. 143-148. (Russian)
[6] I.T. Pirmamedov, Investigating parametric vibrations of nonlinear and thickness inhomogeneous visco-elastic filled cylindric shell by using the Pasternak model//Vestnik Bakinskogo Universiteta, Ser. Phys. Mat. 2 (2005), pp. 93-99. (Russian)
[7] I.T. Pirmamedov, Parametric vibrations of nonlinear and thickness inhomogeneous visco-elastic medium-contacting cylindric shell by using the Pasternak’s dynamical model, Vestnik Kavkazskogo Mezhdun. Universiteta, Tbilisi, Georgia 1 (2009) 20-25. (Russian)
[8] M.A. Mehdiyev, Nonlinear vibrations of stiffened cylindrical shell with a viscoelastic filler, Functional analysis and its applications, in: Proc. of the Int. Conf. devoted to the 100-th anniv. of academician Z.I. Khalilov. Baku, 2011, pp. 245-249(Russian).
[9] M.A. Mehdiyev, Nonlinear parametric vibrations of medium-contacting cylindrical shell with a longitudinally ribs, Mechanics Machine Building 1 (2001) 23-27(Russian).
[10] M.A. Mehdiyev, Nonlinear parametric vibrations of stiffened cylindrical shell with a viscoelastic filler, Mechanics of Machine, Mechanisms and Materials 16 (3)(2011) 28-30. (Russian)
[11] M.A. Mehdiyev, Nonlinear parametric vibrations of viscoelastic medium contacting cylindrical shell stiffened with longitudinal ribs with regard to lateral shift, in: Int. conf. dedicated to 70th Anniv. of the Georgian NAS and the 120th Birthday of its First President Academician N. Muskhelishvili. Batumi, Georgia, 2011, pp. 143-144.
[12] F. Latifov, M. Mehdiyev, Nonlinear parametric vibrations of viscoelastic medium contacting cylindrical shell stiffened with annular ribs and with respect lateral shift, in: Int. conf. Continuum Mechanics and Related Problems of Analysis to Cel, the 70th Anniv. of the Georgian NAS and the 120th Birthday of its First President Academician N. Muskhelishvili, 2011, p. 77.
[13] A.S. Volmir, Nonlinear Dynamics of Plates and Shells, Moscow, Nauka, 1972, p. 432. (Russian).
Institute of Mathematics and Mechanics of NAS of Azerbaijan
Received: December 30, 2011 / Accepted: January 31, 2012 / Published: February 15, 2012.
Abstract: In the paper, a problem on parametric vibrations of a cylindrical shell contacting with external visco-elastic medium, stiffened by a longitudinally ribs and situated under the action of external pressure, is solved in a geometric nonlinear statement by means of the variation principle. Lateral shift of the shell is taken into account. Influences of environment have been taken into account by means of the Pasternak model. The curve separating the stability and instability domains of parametric vibrations has been constructed on the plane ?load-frequency?.
Key words: Ribbed shell, lateral shift, variation principle, nonlinear parametric vibrations.
1. Introduction??
Thin-shelled structural elements of envelope type compose a vast class of mechanical objects that are widely used in contemporary mechanical engineering, in space and rocket technology, and in construction. Strength analysis, stability and vibrations of such constrictions play an essential part in their designing. Nevertheless the behavior of thin-shelled constructions containing ribs that could take into account the discreteness of arrangement of ribs, shift and tensional rigidity of ribs, lateral shifts and geometric nonlinearity has not been studied enough. The reason for this is the complexity of the mentioned factors and necessity of solving bulky nonlinear boundary value problems. Furthermore, some constructions preserve carrying capacity after local loss of stability, and discovery of different forms of stability loss give rise to mathematical complexities. Therefore, development of mathematical models for investigating the behavior of stiffened shells that more completely take into account their work under dynamical loads, carrying out investigations of stability, and vibrations on their base, and also choice of rational parameters of a medium-contacting construction are urgent problems. The solution of such type problems represents some mathematical difficulty that becomes intensified both with regard to dynamical effects and influence of environment, where the elaboration of the approximate method is required. The variational method is one of them.
Ref. [1] has been devoted to the investigation of nonlinear deformation of cylindrical shells under the action of different type dynamical loads. Refs. [2-4] have been devoted to the investigation of stability, vibration and optimization of ridge cylindrical shells. The stability of cylindrical shells of step-variable thickness under the action of dynamical loads has been investigated by the variational-parametric method in Ref. [5].
The problems on parametric vibrations of a nonlinear and thickness inhomogeneous viscoelastic unstiffened filled cylindrical shell have been studied in Refs. [6, 7] by using the variational principle and the Pasternak model. Nonlinear vibrations of a stiffened, viscoelastic medium-contacting cylindrical shell have been researched in geometrically nonlinear statement by using the variational principle in Refs. [8-12].
In the present paper, by means of the variational principle, in geometrically nonlinear statement, for the first time we solve a problem on parametric vibrations of a longitudinally stiffened, viscoelastic medium-filled cylindrical shell subjected to the action of external pressure. The lateral shift of the shell is taken into account. The influence of environment is taken into account with the help of the Pasternak model. On the plane “load-frequency”, the curves separating the stability and instability areas of parametric vibrations are investigated. Influences of lateral shift on critical load parameter of shell’s stability are studied.
2. Problem Statement
We’ll obtain differential equations of motion and natural boundary conditions for a medium-contacting cylindrical shell longitudinally stiffened with regard to lateral shift on the base of Ostrogradsky-Hamilton variational principle. For applying the mentioned principle, we write beforehand the potential and kinetic energy of the system.
The potential energy of elastic deformation of the cylindrical shell with regard to lateral shift is of the form [13]:
(1)
The expressions for the potential energy of elastic deformation of the ?ith longitudinal rib are the followings [2]:
In Eqs. (1) and (2) x1, x2, y1, y2 are the coordinates of curvilinear and linear edges of the shell; Fi, Jzi, Jyi, Jkpi are the area and inertia moments of the cross section of the i-th longitudinal bar with respect to the axis Oz and the axis parallel to the axis Oy and passing through the gravity center of the cross section, and also its inertia moment under torsion; Ei, Gi are the elasticity and shift module of the material of the i-th longitudinal bar.
The potential energy of the shell, under the action of external surface and edge loads applied to the shell is determined as the operation performed by the loads when taking the system from the deformed state to the initial undeformed state, and is represented as follows:
(3)
The potential energy of external edge loads applied to the ends of the i-th longitudinal bar is similarly determined by the following expressions (it is assumed that only boundary loads are applied to the ribs):
??
(4)
The total potential energy of the system equals the sum of potential energies of elastic deformations of the shells and ribs, and also potential energies of all external loads:
A (5)
The kinetic energy of the shell and ribs is written in the form
? are densities of materials from which the shell and the i-th longitudinal bar has been made.
The kinetic energy of the ridge shell is defined as follows:
?
(8)
The equations of motion of a medium-contacting stiffened ridge shell are obtained on the base of Ostrogradsky-Hamilton principle on stationarity of the action: 0
W is the Hamilton action, ???KL is Lagrange’s function, t’ and t” are the given arbitrary times.
We can write the load intensity acting on the shell as viewed from the visco-elastic filler, in the following kc
(Pasternak model), where ▽2 is two-dimensional Laplace’s operator on the contact surface, w is the shell’s deflection, q0, q1 are constants.
Taking the equalities into account the constancy of radial deflections along the height of cross sections and the equality of corresponding twisting angels following from the conditions of rigid joint of ribs with the shell, we write the following relations:
??????;,,1ii
?, are the turning and twisting angles of cross-sections of longitudinal bars.
Allowing for Eq. (11), we express the displacement of bars by the displacement of the shell. From the stationarity Eq. (9) we get nonlinear algebraic equations with respect to desired unknowns.
3. Problem Solution
On an example consider nonlinear parametric vibrations of a longitudinally stiffened annular cylindrical shell with regard to lateral shift and under the action of radial load ,sin110tqqq??? where
? is the pressure change frequency of visco-elastic medium-filled shell. Assuming that the edges of the shell is hingely supported for lx; 0?
;
Where m is the number of waves in the peripheral direction, ??is the frequency of vibration of desired quantities:.
??tt integrate with respect to x, y and t. Then instead of Eq. (5) we get a function from the desired quantities,
4. Conclusions
The dependences of dynamical stability on the parameters of the construction on the plane“load-frequency” represented in the form of a curve are given in Fig. 1. This curve divides the plane into two domains: for the points of one domain the vibrations are restricted, for another one they are unbounded in time. In the graph, the vibrations of longitudinally stiffened cylindrical shell in visco-elastic medium are given by prime lines, by solid lines in elastic medium. Furthermore curve 1 corresponds to discount of lateral shift, in the shell, curve 2 to ignoring lateral shift in the shell. It follows from the calculation results that for a visco-elastic body, the breaking point of the typical curve rises over the frequency axis. Discount of influence of medium reduces to increase of stable zones of the shell, discount of lateral shift in the shell reduces to contraction of stable zones of the shell.
References
[1] V.D. Kubenko, P.S. Kovalchuk, N.P. Podchasov, Nonlinear vibrations of cylindrical shells, Vyshcha shkola, Kiev, 1989, p. 208. (in Russian)
[2] I.Y. Amiro, V.A. Zarutsky, Theory of ribbed shells. Methods of calculating shells, Naukova Dumka, Kiev, 1980, p. 367. (Russian)
[3] I.Y. Amiro, V.A. Zarutsky, V.N. Revutsky, Vibrations of RIbbed shells of Revolution, Naukova Dumka, Kiev, 1988, p. 171.(Russian)
[4] V.A. Zarutsky, Y.M. Pochtman, V.V. Skalozub, Optimization of Reinforced of Cylindrical Shells, Vyshcha Shkola, Kiev, 1990, p. 138. (Russian)
[5] D.I. Aristov, V.V. Karpov, A.Y. Salnikov, Variation-parametric method of investigation of cylindrical shells stepwise variable thickness under dynamic loadingiffened, Mathematical modeling, numerical methods and programs Intercollege Thematic, Sat trudy SPbGASU-SPb, 2004, pp. 143-148. (Russian)
[6] I.T. Pirmamedov, Investigating parametric vibrations of nonlinear and thickness inhomogeneous visco-elastic filled cylindric shell by using the Pasternak model//Vestnik Bakinskogo Universiteta, Ser. Phys. Mat. 2 (2005), pp. 93-99. (Russian)
[7] I.T. Pirmamedov, Parametric vibrations of nonlinear and thickness inhomogeneous visco-elastic medium-contacting cylindric shell by using the Pasternak’s dynamical model, Vestnik Kavkazskogo Mezhdun. Universiteta, Tbilisi, Georgia 1 (2009) 20-25. (Russian)
[8] M.A. Mehdiyev, Nonlinear vibrations of stiffened cylindrical shell with a viscoelastic filler, Functional analysis and its applications, in: Proc. of the Int. Conf. devoted to the 100-th anniv. of academician Z.I. Khalilov. Baku, 2011, pp. 245-249(Russian).
[9] M.A. Mehdiyev, Nonlinear parametric vibrations of medium-contacting cylindrical shell with a longitudinally ribs, Mechanics Machine Building 1 (2001) 23-27(Russian).
[10] M.A. Mehdiyev, Nonlinear parametric vibrations of stiffened cylindrical shell with a viscoelastic filler, Mechanics of Machine, Mechanisms and Materials 16 (3)(2011) 28-30. (Russian)
[11] M.A. Mehdiyev, Nonlinear parametric vibrations of viscoelastic medium contacting cylindrical shell stiffened with longitudinal ribs with regard to lateral shift, in: Int. conf. dedicated to 70th Anniv. of the Georgian NAS and the 120th Birthday of its First President Academician N. Muskhelishvili. Batumi, Georgia, 2011, pp. 143-144.
[12] F. Latifov, M. Mehdiyev, Nonlinear parametric vibrations of viscoelastic medium contacting cylindrical shell stiffened with annular ribs and with respect lateral shift, in: Int. conf. Continuum Mechanics and Related Problems of Analysis to Cel, the 70th Anniv. of the Georgian NAS and the 120th Birthday of its First President Academician N. Muskhelishvili, 2011, p. 77.
[13] A.S. Volmir, Nonlinear Dynamics of Plates and Shells, Moscow, Nauka, 1972, p. 432. (Russian).