Horst Ecker

Vibration suppression and energy transfer by parametric excitation in drive systems

Drive systems may experience torsional vibrations due to various kinds of excitation mechanisms. In many engineering systems, however, such vibrations may have a negative impact on the performance and must be avoided or reduced to an acceptable level by all means. Self-excited vibrations are especially unwanted, since they may grow rapidly and not only degrade the performance but even damage machinery. In this contribution it is suggested to employ parametric stiffness excitation to suppress self-excited vibrations. In the first part of the article we study the basic energy transfer mechanism that is initiated by parametric excitation, and some general conclusions are drawn. In the second part, a hypothetic drivetrain, consisting of an electrical motor, a drive shaft and working rolls is investigated. A self-excitation mechanism is assumed to destabilize the drive system. Parametric excitation is introduced via the speed control of the electrical drive, and the capability of stabilizing the system by this measure is investigated. It is shown that the damping available in the system can be used much more effectively if parametric stiffness excitation is employed.

Stability Analysis of Open-loop Stiffness Control to Suppress Self-excited Vibrations

We present stability investigations on vibration cancelling employing three different types of variable-stiffness actuators. A two-mass system is considered, with a base mass attached to the ground and a top mass connected to the base mass. The top mass is subject to self-excitation forces. The stiffness of an actuator connecting the base mass and the ground may change with time, according to a predetermined control frequency, for cancelling vibrations. Numerical simulation is employed as the basic tool to investigate the system and to carry out parameter studies. The stability of the system is determined by calculating the eigenvalues of the state transition matrix. Robustness of the proposed methods for vibration cancelling is discussed with respect to various aspects.

Design of an electromagnetic actuator for parametric stiffness excitation

Purpose – An overview is given on design features, numerical modelling and testing of a novel electromagnetic actuator to achieve a controllable stiffness to be used as a device for parametric stiffness excitation.Design/methodology/approach – In principle, the actuator consists of a current driven coil placed between two permanent magnets. Repellent forces are generated between the coil and the magnets, centering the coil between the two magnets. The 2D finite element analyses are carried out to predict the forces generated by this arrangement depending on coil current and coil position. Force measurements are also made using the actual device.Findings – Actuator forces as predicted by the finite element analyses are in excellent agreement with the measured data, confirming the validity of the numerical model. Stiffness of the actuator is defined as the increase of force per unit of coil displacement. Actuator stiffness depends linearly on the coil current but in a nonlinear manner on the coil displaceme...

Experimental Study on Cancelling Self-Excited Vibrations by Parametric Excitation

This article reports on the experimental verification of an anti-resonance effect obtained by parametric stiffness excitation. From theoretical studies it is known that parametric excitation at non-resonant parametric resonances can improve the damping behavior of a mechanical system and even stabilize an otherwise unstable system. To demonstrate this effect, a test setup was designed, based on a two-mass vibration system, gliding on an air track. Parametric stiffness excitation (PSE) was realized by a mechanical device that creates a time-periodic stiffness by modulating the tension in an elastic rubber band. With this device it was possible to demonstrate the improved damping behavior of the system when the PSE device is operating at or near the first parametric combination resonance of difference type. Also, a simple electro-magnetic device was used to create self-exciting forces. It could be shown for the first time that it is indeed possible to stabilize the unstable system by introducing parametric stiffness excitation.Copyright

Parametric Excitation in a Two Degree of Freedom MEMS System

This contribution investigates the influence of parametric excitation on the dynamic stability of a microelectrome- chanical system. In systems with just a single degree of freedom, parametric excitation causes the oscillator to exhibit unstable behavior within certain intervals of the parametric excitation frequency. In multi-degree of freedom systems on the other hand, unstable behavior is caused within a wider range of intervals of the parametric excitation frequency. Moreover, such systems show frequency intervals of enhanced stability, an effect known as anti-resonance phenomenon. Both types of phenomena, the parametric resonance and anti-resonance, are modeled and studied for a microelectromechanical system with two degrees of freedom and some novel results are discussed.

Nonlinear stability analysis of a single mass rotor contacting a rigid backup bearing

This paper investigates the steady-state response of a rigid, single mass rotor with imbalance eccentricity supported by an active magnetic bearing with nonlinear characteristics. The rotor may have intermittent contact with an axially collocated, fixed, rigid and circular backup bearing. A radial offset position of the backup bearing center with respect to the magnetic bearing center is assumed. Parameter studies are carried out, especially for the excitation frequency and the friction conditions at the contact point. For frequencies ranging from the onset of contact up to the critical speed various kinds of periodic, non-periodic and quasi-periodic solutions can be observed. Within the parameter range investigated, a two-periodic orbit with one contact was found to be the dominant stable orbit for low excitation frequencies.

Active Damping of Vibrations of a Cantilever Beam by Axial Force Control

In several fields, e.g. aerospace applications, robotics or the bladings of turbomachinery, the active damping of vibrations of slender beams which are subject to free bending vibrations becomes more and more important. In this contribution a slender cantilever beam loaded with a controlled force at its tip, which always points to the clamping point of the beam, is treated. The equations of motion are obtained using the Bernoulli-Euler beam theory and d’Alemberts principle. To introduce artificial damping to the lateral vibrations of the beam, the force at the tip of the beam has to be controlled in a proper way. Two different methods are compared. One concept is the closed-loop control of the force. In this case a nonlinear feedback control law is used, based on axial velocity feedback of the tip of the beam and a state-dependent amplification. By contrast, the concept of open-loop parametric control works without any feedback of the actual vibrations of the mechanical structure. This approach applies the force as harmonic function of time with constant amplitude and frequency. Numerical results are carried out to compare and to demonstrate the effectiveness of both methods.Copyright

A Parametric Absorber for Friction-Induced Vibrations

This contribution deals with the suppression of friction-induced vibrations of a mechanical system. A two-mass system is considered, with the main mass excited by a friction-generated self-excitation force and a smaller second mass attached to the main mass. The parameter of the connecting stiffness between the main mass and the absorber mass is a harmonic function of time and represents a parametric excitation. The purpose of the second mass is to act as a “parametric absorber” and to cancel vibrations. Critical values for the damping parameters of the conventional system are calculated, where the system operates on the stability limit. Analytical and numerical methods are employed to determine the stability of the parameter-excited system. A study for selected parameters shows within which limits friction-induced vibrations can be suppressed effectively by a parametric absorber.Copyright

Stability analysis of a time-periodic 2-dof MEMS structure

Microelectromechanical systems (MEMS) are becoming important for all kinds of industrial applications. Among them are filters in communication devices, due to the growing demand for efficient and accurate filtering of signals. In recent developments single degree of freedom (1-dof) oscillators, that are operated at a parametric resonances, are employed for such tasks. Typically vibration damping is low in such MEM systems. While parametric excitation (PE) is used so far to take advantage of a parametric resonance, this contribution suggests to also exploit parametric anti-resonances in order to improve the damping behavior of such systems. Modeling aspects of a 2-dof MEM system and first results of the analysis of the non-linear and the linearized system are the focus of this paper. In principle the investigated system is an oscillating mechanical system with two degrees of freedom x = [x1x2]T that can be described by Mx+Cx+K1x+K3(x2)x+Fes(x,V(t)) = 0. The system is inherently non-linear because of the cu...

Observations Regarding Numerical Results Obtained by the Floquet-Method

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem.As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods.Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly.Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.Copyright