Communications in Nonlinear Science and Numerical Simulation, 2017
In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the aco... more In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the acoustic metamaterial is investigated. The excitation of the system is in the form of the Jacobi elliptic function. The corresponding model to this forcing is the mass-in-mass system with cubic nonlinearity of the Duffing type. Mathematical model of the motion is a system of two coupled strong nonlinear and nonhomogeneous second order differential equations. Particular solution to the system is obtained. The analytical solution of the problem is based on the simple and double integral of the cosine Jacobi function. In the paper the integrals are given in the form of series of trigonometric functions. These results are new one. After some modification the simplified solution in the first approximation is obtained. The result is convenient for discussion. Conditions for elimination of the motion of the mass 1 by connection of the nonlinear dynamic absorber (mass-spring system) are defined. In the consideration the effective mass ratio is introduced in the nonlinear mass-in-mass system. Negative effective mass ratio gives the absorption of vibrations with certain frequencies. The advantage of the nonlinear subunit in comparison to the linear one is that the frequency gap is significantly wider. Nevertheless, it has to be mentioned that the amplitude of vibration differs from zero for a small value. In the paper the analytical results are compared with numerical one and are in agreement.
This article provides an overview of the exact closed-form solutions for purely nonlinear oscilla... more This article provides an overview of the exact closed-form solutions for purely nonlinear oscillators. These solutions comprise the period of vibration and free and forced responses for one-degree-of-freedom systems. The use of special function for this purpose is demonstrated which includes: beta function, gamma function, hypergeometric function, Ateb function, Jacobi amplitude and Jacobi elliptic function.
Recently, a significant attention has been directed toward so called 'acoustic metamaterials' whi... more Recently, a significant attention has been directed toward so called 'acoustic metamaterials' which have large similarity with already-known 'electromagnetic metamaterials' which are applied for elimination of the electromagnetic waves. The stop of electromagnetic waves is realized with the negative refractive index, negative permittivity and negative permeability. Motivated by the mathematical analogy between acoustic and electromagnetic waves, the acoustic metamaterials are introduced. It was asked the material to have negative effective mass. To obtain the negative effective mass, the artificial material, usually composite, has to be designed. The basic unit is a vibration absorber which consists of a lumped mass attached with a spring to the basic mechanical system. The purpose of the unit is to give a band gap where some frequencies of acoustic wave are stopped. We investigated the nonlinear mass-in-mass unit excited with any periodic force. Mathematical model of the motion is a system of two coupled strongly nonlinear and nonhomogeneous second-order differential equations. The solution of equations is assumed in the form of the Ateb (inverse beta) periodic function. The frequency
In this paper the adiabatic invariants for strongly nonlinear dynamical systems with two degrees ... more In this paper the adiabatic invariants for strongly nonlinear dynamical systems with two degrees of freedom described by complex functions are obtained. The method [1] developed for dynamical systems with one degree of freedom is extended to systems with two degrees of freedom. The method is based on Noether’s theory and the use of Krylov-Bogolubov-Mitropolski (KBM) and elliptic-Krylov-Bogolubov (EKB) asymptotic techniques. The adiabatic invariants for two types of strong nonlinearities are constructed: the pure cubic nonlinearity and quasi-cubic nonlinearity. The adiabatic invariants are used to obtain the approximate solution to the equations of motion.
International Journal of Nonlinear Sciences and Numerical Simulation, 2009
In this paper various analytical asymptotic techniques for solving the strictly strong non-linear... more In this paper various analytical asymptotic techniques for solving the strictly strong non-linear Duffing equation are investigated. The basic function used in the methods is the Jacobi elliptic one. The following methods are emphasized: (1) the elliptic harmonic balance method, (2) the elliptic Galerkin method (the weighted residual method), (3) the straightforward expansion method, (4) the elliptic Lindstedt-Poincare method (parameter-expanding method), (5) the elliptic Krylov-Bogolubov method (the parameter perturbation method), (6) homotopy perturbation method and (7) homotopy analysis method. The methods are tested on the Duffing equation which contains the additional quadratic non-linear term. The obtained approximate analytical solutions are compared with each other and with numerical 'exact' ones. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters. Besides, the methods give much accurate solutions in comparison to the previous one based on the trigonometric functions.
In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Bot... more In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.
In this paper a linguistic multi-criteria method is applied for making decisions in a water plant... more In this paper a linguistic multi-criteria method is applied for making decisions in a water plant to provide the safety operation. If the vibration level of pumps is higher than the limit values prescribed by corresponding standards and, besides, the vibration has the tendency to increase, it is necessary to act, but not to perturb the water supply. Alternatives for acting are: to substitute the pump with a new one, to change the bearings, balancing of the pump's rotor, to rearrange the pumping regime, to fasten the pump construction or to intensify vibration and temperature monitoring of the layers. Which of these actions will be applied depends on the criteria of operation safety, safety of operators, increase of production efficiency, cost of action, in which season of year had happened, is the action necessary and it is necessary to be done instantly. Opinion of experts in field is usually different. The aim of the paper is to give a method for giving the optimal decision due to the mentioned criteria. For these purposes the SofverLingV is developed based on linguistic information. The obtained results agree with those obtained with Software VIKOR which was based on numeric description of the problem by experts. The suggested method is very convenient for application.
The paper is a review on the literature dealing with the main properties of non-ideal vibrating s... more The paper is a review on the literature dealing with the main properties of non-ideal vibrating systems. The analytical and numerical methods applied for analysing such systems are shown. The practical examples of non-ideal systems are considered. The most common phenomenon for the systems are discussed. The specific properties for various models are also discussed. The direction of the future investigation are given.
In this paper a generalization of a two-mass oscillatory system is done. Connection between two m... more In this paper a generalization of a two-mass oscillatory system is done. Connection between two masses is in general of the visco-elastic type where the elastic and damping properties are of the nonlinear type. Motion of the system is described with a system of two coupled second order differential equations (TDE) where the nonlinearity is of any order (integer and/or non-integer). An approximate solution of the TDE is obtained by introducing the intermediate variables which give a single uncoupled differential equation for which the solution is already known. Cveticanin's solution procedure developed for a single second order nonlinear differential equation is extended for solving the TDE. The procedure suggested in the paper is based on the exact analytically determined frequency and period of vibration. The obtained solutions show that for the case of the pure elastic connection the masses oscillate around the same position which is the averaged value of the initial deflections. For both masses the amplitudes and periods of vibration are equal but their motion is in opposite directions. The amplitude of vibration is a linear and frequency a nonlinear (of integer or non-integer order) function of the difference between initial deflections of masses. If in the system damping acts amplitudes of vibration for both masses decrease. The amplitude decrease depends not only on the coefficient of damping (as it was the case for linear systems) but on the initial properties of the system, coefficients of elasticity and order of nonlinearities of the elastic and damping forces. The periods of vibration increase if the damping acts. The frequency of vibration is a complex function of initial displacements of masses, coefficients of elasticity and damping and of the order of nonlinearities of the connection. Two numerical examples illustrate the suggested procedure and results.
In this paper, the dynamics of a cutting mechanism, which is a special type of the crank-shaper m... more In this paper, the dynamics of a cutting mechanism, which is a special type of the crank-shaper mechanism, is analyzed. The influence of the cutting force on the motion of the mechanism is considered. The critical value of the cutting force for which the regular rotating motion of the leading element is disturbed, is denoted. This force corresponds to the buckling force. The buckling force is the limit value for the full continuous rotation of the leading element of the cutting mechanism with the constant moment. The cutting mechanism with special values of parameters is also considered. For such a mechanism the stability conditions and the boundedness in the sense of Lagrange are determined. As a result, the limit value of the cutting force for which the motion is stable is obtained. This value is compared with the value of the dynamic buckling force. In this paper, we have determined the boundary values of the cutting force for which the cutting process is correct. These values are obtained analytically and numerically. Finally, the results are compared.
In the present paper the motion of a lifting mechanism of crane is considered. The unloading proc... more In the present paper the motion of a lifting mechanism of crane is considered. The unloading procedure is analyzed when the mass of the mechanism is varying. Due to the mass variation a reactive force appears. The influence of the reactive force on the motion of the system is investigated. The mechanical model of the mechanism is a simple pendulum with the variable mass and length. Some special cases are considered: (i) the relative mass variation rate is constant; (ii) the damping is varying and the relative length variation rate is constant and the wind force is present. The values of mechanism parameters, for which beside the regular also nonregular motion may appear, are obtained. The method of Melnikov is applied.
In this paper the stability of rotation of the rotor of a textile machine is analysed. The stabil... more In this paper the stability of rotation of the rotor of a textile machine is analysed. The stability is analysed by the use of the methods of Lyapunov. The conditions and parameters of stable and asymptotic stable rotation are denoted for the rotor with disbalance force, for the rotor without reactive force and for the rotor with small non-linear elastic force. At the end a numerical example is analysed.
This paper deals with the longitudinal vibrations of a nonlinear rod. The nonlinearity is strong ... more This paper deals with the longitudinal vibrations of a nonlinear rod. The nonlinearity is strong and of cubic type. The motion of the rod is described by a second-order strong nonlinear partial differential equation. The exact numerical solution is obtained. Four approximate analytical methods for solving the differential equation are developed: the eigenmode solutions are denoted by applying the known space or time distribution functions, respectively, the method based on invariant manifolds, and the method for optimizing the two-mode eigenfrequency solution. The analytical and numerical solutions are compared. The amplitude-time-position diagrams and amplitude-frequency diagrams are plotted.
Communications in Nonlinear Science and Numerical Simulation, 2017
In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the aco... more In this paper the dynamics of the nonlinear mass-in-mass system as the basic subsystem of the acoustic metamaterial is investigated. The excitation of the system is in the form of the Jacobi elliptic function. The corresponding model to this forcing is the mass-in-mass system with cubic nonlinearity of the Duffing type. Mathematical model of the motion is a system of two coupled strong nonlinear and nonhomogeneous second order differential equations. Particular solution to the system is obtained. The analytical solution of the problem is based on the simple and double integral of the cosine Jacobi function. In the paper the integrals are given in the form of series of trigonometric functions. These results are new one. After some modification the simplified solution in the first approximation is obtained. The result is convenient for discussion. Conditions for elimination of the motion of the mass 1 by connection of the nonlinear dynamic absorber (mass-spring system) are defined. In the consideration the effective mass ratio is introduced in the nonlinear mass-in-mass system. Negative effective mass ratio gives the absorption of vibrations with certain frequencies. The advantage of the nonlinear subunit in comparison to the linear one is that the frequency gap is significantly wider. Nevertheless, it has to be mentioned that the amplitude of vibration differs from zero for a small value. In the paper the analytical results are compared with numerical one and are in agreement.
This article provides an overview of the exact closed-form solutions for purely nonlinear oscilla... more This article provides an overview of the exact closed-form solutions for purely nonlinear oscillators. These solutions comprise the period of vibration and free and forced responses for one-degree-of-freedom systems. The use of special function for this purpose is demonstrated which includes: beta function, gamma function, hypergeometric function, Ateb function, Jacobi amplitude and Jacobi elliptic function.
Recently, a significant attention has been directed toward so called 'acoustic metamaterials' whi... more Recently, a significant attention has been directed toward so called 'acoustic metamaterials' which have large similarity with already-known 'electromagnetic metamaterials' which are applied for elimination of the electromagnetic waves. The stop of electromagnetic waves is realized with the negative refractive index, negative permittivity and negative permeability. Motivated by the mathematical analogy between acoustic and electromagnetic waves, the acoustic metamaterials are introduced. It was asked the material to have negative effective mass. To obtain the negative effective mass, the artificial material, usually composite, has to be designed. The basic unit is a vibration absorber which consists of a lumped mass attached with a spring to the basic mechanical system. The purpose of the unit is to give a band gap where some frequencies of acoustic wave are stopped. We investigated the nonlinear mass-in-mass unit excited with any periodic force. Mathematical model of the motion is a system of two coupled strongly nonlinear and nonhomogeneous second-order differential equations. The solution of equations is assumed in the form of the Ateb (inverse beta) periodic function. The frequency
In this paper the adiabatic invariants for strongly nonlinear dynamical systems with two degrees ... more In this paper the adiabatic invariants for strongly nonlinear dynamical systems with two degrees of freedom described by complex functions are obtained. The method [1] developed for dynamical systems with one degree of freedom is extended to systems with two degrees of freedom. The method is based on Noether’s theory and the use of Krylov-Bogolubov-Mitropolski (KBM) and elliptic-Krylov-Bogolubov (EKB) asymptotic techniques. The adiabatic invariants for two types of strong nonlinearities are constructed: the pure cubic nonlinearity and quasi-cubic nonlinearity. The adiabatic invariants are used to obtain the approximate solution to the equations of motion.
International Journal of Nonlinear Sciences and Numerical Simulation, 2009
In this paper various analytical asymptotic techniques for solving the strictly strong non-linear... more In this paper various analytical asymptotic techniques for solving the strictly strong non-linear Duffing equation are investigated. The basic function used in the methods is the Jacobi elliptic one. The following methods are emphasized: (1) the elliptic harmonic balance method, (2) the elliptic Galerkin method (the weighted residual method), (3) the straightforward expansion method, (4) the elliptic Lindstedt-Poincare method (parameter-expanding method), (5) the elliptic Krylov-Bogolubov method (the parameter perturbation method), (6) homotopy perturbation method and (7) homotopy analysis method. The methods are tested on the Duffing equation which contains the additional quadratic non-linear term. The obtained approximate analytical solutions are compared with each other and with numerical 'exact' ones. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters. Besides, the methods give much accurate solutions in comparison to the previous one based on the trigonometric functions.
In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Bot... more In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.
In this paper a linguistic multi-criteria method is applied for making decisions in a water plant... more In this paper a linguistic multi-criteria method is applied for making decisions in a water plant to provide the safety operation. If the vibration level of pumps is higher than the limit values prescribed by corresponding standards and, besides, the vibration has the tendency to increase, it is necessary to act, but not to perturb the water supply. Alternatives for acting are: to substitute the pump with a new one, to change the bearings, balancing of the pump's rotor, to rearrange the pumping regime, to fasten the pump construction or to intensify vibration and temperature monitoring of the layers. Which of these actions will be applied depends on the criteria of operation safety, safety of operators, increase of production efficiency, cost of action, in which season of year had happened, is the action necessary and it is necessary to be done instantly. Opinion of experts in field is usually different. The aim of the paper is to give a method for giving the optimal decision due to the mentioned criteria. For these purposes the SofverLingV is developed based on linguistic information. The obtained results agree with those obtained with Software VIKOR which was based on numeric description of the problem by experts. The suggested method is very convenient for application.
The paper is a review on the literature dealing with the main properties of non-ideal vibrating s... more The paper is a review on the literature dealing with the main properties of non-ideal vibrating systems. The analytical and numerical methods applied for analysing such systems are shown. The practical examples of non-ideal systems are considered. The most common phenomenon for the systems are discussed. The specific properties for various models are also discussed. The direction of the future investigation are given.
In this paper a generalization of a two-mass oscillatory system is done. Connection between two m... more In this paper a generalization of a two-mass oscillatory system is done. Connection between two masses is in general of the visco-elastic type where the elastic and damping properties are of the nonlinear type. Motion of the system is described with a system of two coupled second order differential equations (TDE) where the nonlinearity is of any order (integer and/or non-integer). An approximate solution of the TDE is obtained by introducing the intermediate variables which give a single uncoupled differential equation for which the solution is already known. Cveticanin's solution procedure developed for a single second order nonlinear differential equation is extended for solving the TDE. The procedure suggested in the paper is based on the exact analytically determined frequency and period of vibration. The obtained solutions show that for the case of the pure elastic connection the masses oscillate around the same position which is the averaged value of the initial deflections. For both masses the amplitudes and periods of vibration are equal but their motion is in opposite directions. The amplitude of vibration is a linear and frequency a nonlinear (of integer or non-integer order) function of the difference between initial deflections of masses. If in the system damping acts amplitudes of vibration for both masses decrease. The amplitude decrease depends not only on the coefficient of damping (as it was the case for linear systems) but on the initial properties of the system, coefficients of elasticity and order of nonlinearities of the elastic and damping forces. The periods of vibration increase if the damping acts. The frequency of vibration is a complex function of initial displacements of masses, coefficients of elasticity and damping and of the order of nonlinearities of the connection. Two numerical examples illustrate the suggested procedure and results.
In this paper, the dynamics of a cutting mechanism, which is a special type of the crank-shaper m... more In this paper, the dynamics of a cutting mechanism, which is a special type of the crank-shaper mechanism, is analyzed. The influence of the cutting force on the motion of the mechanism is considered. The critical value of the cutting force for which the regular rotating motion of the leading element is disturbed, is denoted. This force corresponds to the buckling force. The buckling force is the limit value for the full continuous rotation of the leading element of the cutting mechanism with the constant moment. The cutting mechanism with special values of parameters is also considered. For such a mechanism the stability conditions and the boundedness in the sense of Lagrange are determined. As a result, the limit value of the cutting force for which the motion is stable is obtained. This value is compared with the value of the dynamic buckling force. In this paper, we have determined the boundary values of the cutting force for which the cutting process is correct. These values are obtained analytically and numerically. Finally, the results are compared.
In the present paper the motion of a lifting mechanism of crane is considered. The unloading proc... more In the present paper the motion of a lifting mechanism of crane is considered. The unloading procedure is analyzed when the mass of the mechanism is varying. Due to the mass variation a reactive force appears. The influence of the reactive force on the motion of the system is investigated. The mechanical model of the mechanism is a simple pendulum with the variable mass and length. Some special cases are considered: (i) the relative mass variation rate is constant; (ii) the damping is varying and the relative length variation rate is constant and the wind force is present. The values of mechanism parameters, for which beside the regular also nonregular motion may appear, are obtained. The method of Melnikov is applied.
In this paper the stability of rotation of the rotor of a textile machine is analysed. The stabil... more In this paper the stability of rotation of the rotor of a textile machine is analysed. The stability is analysed by the use of the methods of Lyapunov. The conditions and parameters of stable and asymptotic stable rotation are denoted for the rotor with disbalance force, for the rotor without reactive force and for the rotor with small non-linear elastic force. At the end a numerical example is analysed.
This paper deals with the longitudinal vibrations of a nonlinear rod. The nonlinearity is strong ... more This paper deals with the longitudinal vibrations of a nonlinear rod. The nonlinearity is strong and of cubic type. The motion of the rod is described by a second-order strong nonlinear partial differential equation. The exact numerical solution is obtained. Four approximate analytical methods for solving the differential equation are developed: the eigenmode solutions are denoted by applying the known space or time distribution functions, respectively, the method based on invariant manifolds, and the method for optimizing the two-mode eigenfrequency solution. The analytical and numerical solutions are compared. The amplitude-time-position diagrams and amplitude-frequency diagrams are plotted.
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Papers by L. Cveticanin