L. Cveticanin

Chaos in Non-ideal Mechanical System with Clearance

In this paper a non-ideal mechanical system with clearance is considered. The mechanical model of the system is an oscillator connected with an unbalanced motor. Due to the existence of clearance the connecting force between motor and the fixed part of the system is discontinuous but linear. The mathematical model of the system is represented by two coupled second-order differential equations. The transient and steady-state motion and also the stability of the system are analyzed. The Sommerfeld effect is detected. For certain values of the system parameters the motion is chaotic. This is caused by the period doubling bifurcation. The existence of chaos is proved with maximal Lyapunov exponent. A new chaos control method based on the known energy analysis is introduced and the chaotic motion is transformed into a periodic one.

Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type

In this paper, an approximate analytic procedure is developed for solving the strong non-linear differential equations of the Duffing type with complex-valued function which describes the dynamical behavior of many real systems. The method is based on the elliptic-Krylov–Bogolubov procedure where the solutions are the Jacobi elliptic functions. As an example, the self-excited vibrations of the rotor with variable shaft rigidity are considered. The analytical results of this example are compared with numerical ones and excellent agreement is found between them.

Chaotic Responses in a Stable Duffing System of Non-ideal Type

In this paper the dynamics of a non-linear system with non-ideal excitation are studied. An unbalanced motor with a strong non-linear structure is considered. The excitation is of non-ideal type. The model is described with a system of two coupled strong non-linear differential equations. The steady state motions and their stability is studied applying the asymptotic methods. The existence of the Sommerfeld effect in such non-linear non-idealy excited system is proved. For certain values of system parameters chaotic motion appears. The chaos is realized through period doubling bifurcation. The results of numerical simulation are plotted and the Lyapunov exponents are calculated. The Pyragas method for control of chaotic motion is applied. The parameter values for transforming the chaos into periodical motion are obtained.

Vibrations of a textile machine rotor

Abstract In this paper the vibrations of a textile machine rotor, whose angular velocity is constant, are analyzed. The function of the rotor is to wind up a band of textile material into a roll. The elastic force in the shaft is assumed to be non-linear. First the free vibrations of this rotor are analyzed analytically and numerically. The results are compared. After that the vibrations in the non-resonant case are analyzed. The solution is found by use of the analytical method of multiple scales. The results for free vibrations and for the non-resonant case are compared.

Resonant vibrations of nonlinear rotors

Abstract In this paper the primary resonance of a non-linear rotor system under the influence of monofrequency excitation is studied. The nonlinearity is caused by non-linear elastic material properties. The rotor system is a two-degree-freedom system described by a differential equation with complex deflection function. An averaging method for obtaining the amplitude and phase variation in time is developed. It is based on the method of slow varying amplitude and phase developed for one-degree-of-freedom system. The steady state solution is also considered. The method is applied for a rotor with small damping, nonlinearity and excitation function. The transient and steady state motion of the system are analyzed for various values of parameters. An experiment is carried out with shafts made of different materials. The analytical solutions are proved experimentally.

Oscillator with a Sum of Noninteger-Order Nonlinearities

Free and self-excited vibrations of conservative oscillators with polynomial nonlinearity are considered. Mathematical model of the system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the case when only one nonlinear term exists, the exact analytical solution of the differential equation is determined as a cosine-Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non-linear order. Exact solutions of the averaged differential equations of motion are obtained. The obtained results are compared with “exact” numerical solutions and previously obtained analytical approximate ones. Advantages and disadvantages of the suggested procedure are discussed.

Asymptotic methods for vibrations of the pure non-integer order oscillator

In this paper oscillators with a restoring force which is the function of a non-integer power exponent of deflection are considered. The oscillatory motion is described by a differential equation with a rational-power term. The equation is first analyzed qualitatively. The new analytical methods are developed for solving the differential equation with a non-integer order term. The methods are based on the assumption that the vibration of the non-integer oscillator has to be close to that of integer order one. The new perturbation method based on variation of the order of the non-linearity is developed. The unperturbed system is the integer order non-linear oscillator. One of the methods uses the perturbation of the amplitude and phase, and the following two techniques introduce the straightforward expansion using the known values for the pure integer order oscillators. The first order approximate solutions are obtained. Their accuracy is checked on several examples. The results obtained are compared with the exact numerical solution, showing good agreement. The vibrations are widely discussed.

A non-simultaneous variational approach for the oscillators with fractional-order power nonlinearities

Abstract This paper is concerned with a class of conservative oscillators the restitution force of which is of a power form which includes positive non-integer exponents. It is shown how an approximate Lagrangian and Hamilton’s variational principle can be used to obtain a second-order approximate solution for their free vibrations. Due to the fact that, in a general case, when the restoring force is multi-term, the period cannot be obtained from the energy conservation law in a closed form, the problem is formulated as a one-point boundary-value problem, and a non-simultaneous variation is introduced. The explicit expressions for the amplitudes and frequency of oscillations are derived, in which there are no restrictions on the values of the non-integer powers. The analytically obtained results are compared with numerical results as well as with some approximate analytical results from the literature.

On the Dynamics of Bodies With Continual Mass Variation

In this paper the differential equations of the general motion of the rigid body with continual mass variation are considered. The impact force and the impact torque that occur due to addition or separation of the body with velocity and angular velocity which differs from the velocity of mass center and angular velocity of the existing body are introduced. The theoretical consideration is applied for solving a real technical problem when the band winds up on the drum. The plane motion of the drum on which the band winds up is considered. The influence of the velocity of the band on the angular velocity of the drum and the motion of the drum mass center is obtained.

An asymptotic solution for weak nonlinear vibrations of the rotor

Abstract In the paper an asymptotic solution for nonlinear vibrations of the rotor, which is under action of normal and tangential forces, is obtained. The procedure is based on the well-known methods of linear vibrations and the asymptotic method of Bogolubov-Mitropolski. The Bogolubov-Mitropolski method is adopted for a differential equation with complex function and small nonlinearity. As a special case, dynamics of the rotor under influence of hydrodynamic force is analyzed. The comparison between obtained results for the following force types: (a) weak and linear, (b) strong linear and (c) weak nonlinear is given. The influence of nonlinearity is significant.