论文部分内容阅读
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?.
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]:
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.
[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,
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?.
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]:
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.
[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,