Alexander Fidlin

Analytical approximations for stick-slip vibration amplitudes

Abstract The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that first decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick–slip and pure-slip oscillations. For stick–slip oscillations, this is accomplished by using perturbation analysis for the finite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the difference in static and kinetic friction is not too large.

Predicting vibration-induced displacement for a resonant friction slider

Abstract A mathematical model is set up to quantify vibration-induced motions of a slider, sandwiched between friction layers with different coefficients of friction, and equipped with an imbedded resonator that oscillates at high frequency and small amplitude. This model is highly nonlinear, involving non-smooth functions with strong harmonic excitation terms. The method of averaging is extended to hold for systems of this class, and used to derive approximate expressions for predicting average velocities of the slider. These expressions are shown to produce results that agree very well with numerical integration of the full equations of motion. The expressions are used to estimate and explain the influence of system parameters.

On the asymptotic analysis of discontinuous systems

Discontinuous systems are usual in mechanics. Main examples are systems with dry friction and systems with collisions. Peculiarities of these systems make them very difficult for conventional asymptotic analysis developed for continuous systems, i.e. requiring from the right-hand sides of the equations of motions a definite level of continuity. The main idea of this paper is to transform discontinuous systems to special forms being natural for the corresponding problem. Averaging theorems are formulated and proved for these forms of dynamical systems. The approach is illustrated with examples of vibration-induced displacement.

On the separation of motions in systems with a large fast excitation of general form

In this study dynamic systems are considered, in which motion can be described through a system of second-order ordinary differential equations with the right sides depending both on the slow time t and on the fast time τ=ωt (ω≫1 is a big dimensionsless parameter). It is assumed that the right sides are large (they have the magnitude order ω) and depend both on generalised coordinates and on the generalised velocities of the system. A motion separating procedure is developed for the systems described in twi ways. The procedure enables separate systems to derive for fast (oscillating) and slow components of the solution. Each of these separated systems is simpler than the original one. The equivalence of both procedures is shown. The first of them is based on the multiple scales method, the second one generalises the averaging method of Krylov–Bogoliubov–Mitropolskii. Motion of a linear oscillator excited through the large fast oscillations of the damping coefficient is analysed as an example of the established method usage. It is shown, that the excitation significantly changes the effective mass and in consequence the natural frequency of the original system. The analytic results are compared with numerical ones.

Regularization of 2D Frictional Contacts for Rigid Body Dynamics

Classic rigid body mechanics does not provide frictional forces acting in a 2D contact interface between two bodies during sticking. This is due to the statical undeterminacy related to this problem. Many technical systems, e.g. disk clutches, have such surface-to-surface contacts and it is sometimes desirable to treat them as rigid body systems despite the 2D contact. Alternatively, it is possible to model the systems using elastic instead of rigid bodies, but this might lead to certain drawbacks. A new regularization model of such 2D contacts between rigid bodies is proposed. It is based on the similarity to a material model for elasto-plasticity in continuum mechanics. Only dry friction is taken into account.

Simulation of torsional vibration dampers

new method to support the development and optimisation of torsional vibration dampers is presented. All relevant operating points, all parameters critical to function and their production tolerances are taken into consideration. In addition to product optimisation and quality assurance, the systematic use of this method reveals conflicting targets and helps to establish the limits of existing design concepts. This results in a targeted search for new and innovative ideas, as demonstrated here with the example of the centrifugal pendulum-type absorber.

Simultaneous optimization of gait and design parameters for bipedal robots

A walking bipedal robots energy efficiency depends on its gait as well as its design, whereas design changes affect the optimal gaits. We propose a method to take these interdependencies into account via simultaneous optimization of gait as well as design parameters. The method is applied to a planar robot with hybrid zero dynamics control and a torsion spring between its thighs. Periodic gaits are simulated by means of the hybrid zero dynamics. The implementation of the simultaneous optimization of gait parameters and spring stiffness via sequential quadratic programming is presented. Subsequently, an error analysis is performed to gain good convergence and short computation times of the optimization. The evaluation of gradients is identified as crucial for the algorithms convergence and therefore performed via complex step derivative approximations. The resulting implementation exhibits good convergence behavior and is provided as supplement to this paper. At 2.3 m/s, the simultaneous optimization results in savings in energy expenditure of up to 55%. A consecutive optimization of first gait and then stiffness yields only 11%, demonstrating the advantage of the presented method.

Oscillator in a Clearance: Asymptotic Approaches and Nonlinear Effects

Averaging combined with non-smooth unfolding transformations is used in this paper for investigating of the basic properties of an oscillator in a clearance. The classical stereo-mechanical approach is used in order to describe collisions between the mass and the limits. Energy dissipation during collision events is taken into account. The analysis is concentrated on the oscillation regimes with alternating collisions with both sides of the clearance. The self-excited friction oscillator in a clearance is considered as the first example. It is shown that applying the unfolding transformation the oscillator can be converted to an almost conservative pendulum rotating in a limited, non-smooth periodical potential field. Analytic predictions are obtained for the total energy of the pendulum which is the natural measure for the oscillations intensity. It is shown that the oscillation frequency can be controlled by changing the negative slope of the friction characteristics, by alternating the normal force and varying the length of the clearance. The classical externally excited oscillator in the clearance is considered as the second example. Applying the same approach the problem of the resonant oscillations can be reduces to the analysis of the rotation of the harmonically excited pendulum in the same potential field as in the first example. The main attention is paid to the high-energy resonances. The perturbation analysis in the vicinity of the resonant surfaces enables to dissociate slow, semi-slow and fast motions and to obtain very accurate analytic predictions for the energies of stationary resonant regimes. It is demonstrated that different high-energy regimes exist alongside with low-energy oscillations. The last case is typical for relative strong energy losses during collision events, so that the system needs several collision-free oscillations in order to increase the amplitude and reach the limits of the clearance.Copyright

Zur Synthese von Antriebssystemen

Oft ist es wesentlich, bei der Konzeption eines Antriebssystems von Anfang an ein strukturell gunstiges Funktionsprinzip zu finden und die richtige Auswahl unter mehreren geeigneten Baugruppen zu treffen, vgl. dazu Abschn. 5.7. Es sei in diesem Zusammenhang auch erwahnt, dass haufig die wesentlichen konstruktiven Kennzeichen einer antriebstechnischen Losung durch Patente geschutzt sind. So stecken z. B. in der konstruktiven Losung fur schaltbare Kupplungen im Antriebsstrang eines PKW, eines Hybridantriebs oder eines stufenlos verstellbaren Getriebes jeweils hunderte von Patenten! Man kann eigentlich erfinderische Ideen, die definitionsgemas jedes Patent enthalten muss, nicht „ausrechnen“. Oft kann man aber durch die Modellberechnungen die sinnvollen Parametergrenzen des jeweiligen Funktionsprinzips ermitteln und daraufhin sich zu neuen Losungen inspirieren lassen.

Beispiele zur dynamischen Analyse von Antriebssystemen

Bei der Auslegung von Antriebssystemen mit Asynchronmotoren ist die dynamische Analyse mit spezifischen Programmen fur die Simulation zu empfehlen, vgl. Tab. 2.1. Derartige Software stutzt sich u. a. auf mathematische Modelle, die das dynamische Verhalten der Asynchronmotoren beschreiben. Die Dimensionierung der Motoren, Getriebe, Wellen und Kupplungen kann damit unter Berucksichtigung sowohl der technologischen Belastungen als auch der aus den mechanisch-elektrischen Wechselwirkungen innerhalb des Asynchronmotors entstehenden Antriebsmomente ermittelt werden, zu denen die Lastfalle Anlassen, Bremsen, Fehlsynchronisation oder Kurzschluss (2- oder 3-polig) gehoren.