Jaroslav Zapoměl

A computational investigation of the transient response of an unbalanced rigid rotor flexibly supported and damped by short magnetorheological squeeze film dampers

Due to manufacturing and assembly inaccuracies, real rotors are always slightly imbalanced. This produces their lateral vibration and forces that are transmitted through the bearings to the stationary parts. The oscillation of the system can be reduced if damping devices are added to the constraint elements. To achieve the optimum performance of the rotor in a wide range of angular velocities and when passing through the critical speeds their damping effect must be controllable. For this purpose, the application of semiactive magnetorheological (MR) dampers has been analysed. The investigated problem focuses on studying the influence of their damping effect and of its control on the amplitude of the rotor vibration, on the magnitude of the force transmitted to the rotor casing, and on the amount of dissipative power generated in the MR films. The developed mathematical model assumes cavitation in the lubricating layer, and the MR liquid is modelled as a Bingham material. The derivation of the equation governing the pressure distribution in the oil film is completed by a new methodology making it possible to determine the yielding shear stress needed for its solution. The equations of motion of the rotor are nonlinear due to the damping forces and to solve them a Runge–Kutta integration method was applied. Computer simulations show that a suitably proposed current–rotor angular speed relationship enables one to fully eliminate the resonance peaks and to achieve the optimum compromise between the attenuation of the rotor lateral vibration, the magnitude of the forces transmitted to the rotor casing and the amount of energy dissipated in the lubricating layers.

Identification of material damping of a carbon composite bar and study of its effect on attenuation of its transient lateral vibrations

Reduction of noise and vibrations is one of the major requirements put on operation of modern machines. It can be achieved by application of new materials. The ability to utilize them properly requires learning more about their mechanical properties. Vibration attenuation depends on material damping as an important factor. This paper presents the results of research in a carbon composite material focusing on its internal damping, on the measurement of the damping coefficients and on its implementation into mathematical models. The obtained results were used for investigation of suppressing lateral vibrations of a long homogeneous carbon composite bar oscillating in the resonance area. During the transient period and due to nonlinear effects, the harmonic time-varying loading excites the bar response consisting of a number of harmonic components. The specific damping capacity referred to several oscillation frequencies determined by measurement. The results were evaluated from the point of view of two simple damping theories — viscous and hysteretic. The experiments showed that internal damping of the investigated material could be considered as frequency independent. Therefore, in order to carry out simulations, the bar was represented in the computational model by an Euler beam constituted of Maxwell–Weichert theoretical material. A suitable setting of material constants enabled reaching a constant value of the damping parameters in the required frequency range. The investigated bar vibration is governed by the motion equation in which the internal damping forces depend not only on instantaneous magnitudes of the system’s kinematic parameters but also on their past history. Solution of the equations of motion was performed after its transformation into the state space in the time domain. Results of the computational simulations showed that material damping significantly reduced amplitude of the bar vibrations in the resonance area. The producers of composite materials usually provide material parameters allowing to solve various stationary problems (density, modulus of elasticity, yielding point, strength, etc.), but there is only little or almost no information concerning the data needed for carrying out dynamical or other time-dependent analyses such as internal damping coefficients, fatigue limit, etc. Therefore, determination of the hysteretic character of material damping of the investigated carbon composite material, measurement of its specific damping capacity and implementation of the frequency-independent damping into the computational model are the principal contributions of this article.

A Computational Investigation of the Steady State Vibrations of Unbalanced Flexibly Supported Rigid Rotors Damped by Short Magnetorheological Squeeze Film Dampers

Unbalance is the principal cause of excitation of lateral vibrations of rotors and generation of the forces transmitted through the rotor supports to the foundations. These effects can be significantly reduced if damping devices are added to the constraint elements. To achieve their optimum performance, their damping effect must be controllable. The possibility of controlling the damping force is offered by magnetorheological squeeze film dampers. This article presents an original investigation of the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological squeeze film dampers. In the computational model, the rotor is considered as absolutely rigid and the dampers are represented by force couplings. The pressure distribution in the lubricating layer is governed by a modified Reynolds equation adapted for Bingham material, which is used to model the magnetorheological fluid. To obtain the steady state solution of the equations of motion, a collocation method is employed. Stability of the periodic vibrations is evaluated by means of the Floquet theory. The proposed approach to study the behavior of rigid rotors damped by semi-active squeeze film magnetorheological dampers and the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent new contributions of this article.

Vibration attenuation of an electromechanical system coupled with plate springs damped by an impact element

The main aim of this paper is focused on vibration attenuation of the electromechanical system flexibly coupled with a baseplate and damped by an impact element. The model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The system movement is described by three mutually coupled second-order ordinary differential equations, derived by the force balance method. Here, the most important nonlinearities are: stiffness of the support spring elements and internal impacts. The main results show the impact damping device attenuates vibrations of the rotor frame in a wide range of the excitation frequencies, that is wider then in the case when the impact element works only as an inertia damper without occurrence of any impacts.

DYNAMIC PROPERTIES OF THE ELECTROMECHANICAL SYSTEM DAMPED BY AN IMPACT ELEMENT WITH SOFT STOPS

The main aim of this paper is to focus on analysis of the dynamic properties of the electromechanical system with an impact element. This model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational, and is completed with an electric circuit. The mathematical model of the system is represented by three mutually coupled second-order ordinary differential equations. Here, the most important nonlinearities are: stiffness of the support spring elements and internal impacts. Several important results were obtained by means of computational simulations. The impacts considerably increase the number of resonance peaks of the frequency response characteristic. Character of the system motion strongly depends on the width of clearances between the impact body and the rotor frame and changes from simple periodic to close to chaotic or to periodic with a large number of ultraharmonic components. For a suitably chosen system parameters, a significant damping effect of the impact element was observed.

A computational investigation of the disk-housing impacts of accelerating rotors supported by hydrodynamic bearings

As the radial clearance between disks and the casing of rotating machines is usually very narrow, excessive lateral vibration of accelerating rotors passing critical speeds can produce impacts between the disks and the housing. The computer modeling method is an important tool for investigating such events. In the developed procedure, the shaft is flexible and the disks are absolutely rigid. The hydrodynamic bearings and the impacts are implemented in the mathematical model by means of nonlinear force couplings. Most of the publications and computer codes from the field of rotor dynamics are referred only in the case when the rotor turns at a constant angular speed and in simple cases of disk-housing impacts. Moreover, if the disks turning at variable speed are investigated, the resulting form of the equations of motion derived by different authors slightly differs and the differences depend on the method used for their derivation. Therefore, particular emphasis in this article is given to the derivation of the motion equations of a continuous rotor turning with variable revolutions to explain the mentioned differences, to develop a computer algorithm enabling the investigation of cases when impacts between an arbitrary number of disks and the stationary part take place, and to analyze the mutual interaction between the impacts and the fluid film bearings. The Hertz theory is applied to determine the contact forces. Calculation of the hydrodynamic forces acting on the bearings is based on solving the Reynolds equation and taking cavitation into account. Lagrange equations of the second kind and the principle of virtual work are used to derive equations of motion. The Runge―Kutta method with an adaptive time step is applied for their solution. The applicability of the developed procedure was tested by computer simulations. The results show that it can be used for the modeling of complex rotor systems and that the short computational time enables carrying out calculations for a number of design and operation parameters.

ANALYSIS OF THE STEADY STATE UNBALANCE RESPONSE OF RIGID ROTORS ON MAGNETORHEOLOGICAL DAMPERS: STABILITY, FORCE TRANSMISSION AND ENERGY DISSIPATION

The rotors working in real technological devices are always slightly imbalanced. This excites their lateral vibrations and generates forces that are transmitted to the rotor casing. These effects can be significantly reduced if damping devices are added to the support elements. The possibility of controlling the damping, in order to achieve their optimum performance, is offered by magnetorheological squeeze film dampers. In this paper, a computational modeling method is used to investigate the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological dampers. The equations of motion of the rotor are nonlinear due to the damping forces. Computational procedures were developed to verify the applicability of such dampers by simulating their behavior and analyzing their effect on the amplitude of the rotor vibration, on the magnitude of the forces transmitted to the rotor casing and on the amount of the power dissipated in the magnetorheological films. The proposed approach to study the optimum performance of semiactive magnetorheological dampers applied in rotor systems, in terms of vibration amplitudes and transmitted forces, together with the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent the new contributions of this paper.

Stability Investigation of the Steady State Response of Flexibly Supported Rigid Rotors

A new semiactive coupling element working on the principle ofsqueezing two concentric films formed by classical and magnetorheological liquids has been proposed to achieve the optimum attenuation of the lateral vibration of rotors. The damping effect is controlled by changing the magnetic field induction. The steady state solution of the equations of motion is obtained by application of a collocation method and its stability is evaluated using the Floquet theory. The newly proposed controllable damping device enables to minimize amplitude of the rotor vibration by adapting the damping effect to the current operating conditions.

Dynamics and Efficiency of an Impact Damper

The main aim of this paper is to focus on dynamics of the electromechanical system flexibly coupled with a baseplate and damped by an impact element. The model is constructed with three degrees of freedom in the mechanical oscillating part, two translational and one rotational. The system movement is described by three mutually coupled second-order ordinary differential equations. Here, the most important nonlinearities are: stiffness of the support spring elements and internal impacts. As it is shown in the main results, the impact damping device attenuates vibrations of the rotor frame in a wide range of the excitation frequency and the system is showing periodic and chaotic behavior.

Approximate Identification of Dynamical Systems

Usually, as the input data of the parametric identification methods in the frequency domain, the corresponding pairs of the “unit harmonic force excitation”—“steady state harmonic response” are considered. This paper deals with approximate identification of linear dynamical systems by time response on unknown initial displacement (or velocity) with the help of the Fourier transform. In this paper, basic analytical relationships and identification alternatives are analyzed. Formulae are completed and presented with special consideration given to the simplest one mass dynamical system.