V. Ph. Zhuravlev

Theory of the shimmy phenomenon

The shimmy phenomenon is the appearance of angular self-excited vibrations of the carriage wheels. Such self-excited vibrations provide a serious safety hazard for motion, which explains the great interest of scientists in this phenomenon [1–6]. This problem is most serious for the aircraft fore wheels.It is commonly agreed that themain cause of the shimmyphenomenon is the tyre deformation [2]. We do not doubt this thesis, but still we note that this cause is not unique. The shimmy phenomenon can be observed in everyday life in the case of various hand trucks with rigid wheels, where the reference to the elastic tyre is out of place.In what follows, we show that the theory of polycomponent dry friction can completely explain the shimmy phenomenon for absolutely rigid wheels, and hence can be at least one of the causes of this phenomenon in the general case.Dry friction has been ignored by the scientists in their explanations of the shimmy phenomenon, because this friction has not been fully investigated until now, and it has been impossible to explain the shimmy phenomenon in the framework of the former representations.

Global Motion of the Celt

The motion of the celt attracts researchers’ attention by unusual properties. If we put the celt on a horizontal surface and twist it about its vertical axis, then, after a certain time, it stops rotating about this axis, but vibrations about the other axes may arise and then the celt begins rotating about the vertical axis in the opposite direction.Apparently, the first publications about the celt are [1, 2].The known studies of the celt motion contain some simplifying assumptions and usually perform only a local stability analysis of steady-state rotations. Of the numerous publications, we mention [3, 4], where detailed references can be found.We note that the frequently encountered assumption that the velocity of the celt point of contact with the plane is zero (a nonholonomic statement of the problem) has not been confirmed by physical considerations and must be justified separately.In the present paper, we consider the global motion of the celt for the case in which the interaction of the celt with the plane on which it moves is determined by the forces taking into account the mutually depending sliding and twisting, which makes them really physically meaningful [5]. Another example of application of dry friction model used in the present paper can be found in [6].

Effect of movability of the resonator center on the operation of a hemispherical resonator gyro

It was established in [2] that resonator deformation according to the second mode shape of a thin hemispherical shell results in a displacement of the center of mass if the resonator is unbalanced, i.e., if the distribution of mass over the surface of the hemisphere deviates from axial symmetry. In the same paper, it was shown that this displacement of the center of mass makes the instrument sensitive to linear vibrations. The present paper deals with linear vibration caused in the presence of unbalance by the working vibrations themselves and by the forces used to maintain the latter. The linear vibration is considered in the form of beam vibrations of the resonator stem. The study is aimed at determining the influence of the coupling between the working and beam vibrations on the instrument readings. We obtain a formula relating the hemispherical resonator gyro drift to the unbalance and the eccentricity, which, in particular, can be caused by the gravity component normal to the sensitivity axis. The drift considered here is essentially caused by the fact that deformation of the resonator supports also results in deformation of the electric control field in the gap between the electrodes. The resulting additional forces cause the effect studied in this paper. The drift magnitude depends on how the control of the phase state of the resonator is chosen. In what follows, to be definite, we consider the control in fast-time mode, i.e., at the natural vibration frequency. A similar effect takes place for any other type of control of waves in the resonator.

On the history of the dry friction law

We discuss the role played by Amontons, Coulomb, Leonardo daVinci,Morin, and Euler in the discovery of the dry friction law. In particular, reference is given to Euler’s 1748 paper,where an exhaustive statement of the dry friction law was given 37 years before Coulomb’s main publications. Painlevé’s criticism of the Coulomb law is assessed, and the main points of the theory of polycomponent dry friction are given.

Strapdown inertial navigation system of pendulum type

We consider two implementations of an isotropic oscillator to be used in the construction of a strapdown inertial system of new type [1]. The first implementation is a contact-free suspension of an electrically conductive ball used in electrostatic gyros, and the second implementation is a 3D combination of elastic constraints holding a mass point.The inertial system of a new type has the minimum possible dimension. It differs from the already known (platform and strapdown) inertial systems in that it permits one to do without gyros and Poisson integrators because it also performs the functions of an apparent acceleration transducer.

On the precession of the elliptic mode shape of a circular ring owing to nonlinear effects

The Foucault pendulum, which maintains the plane of its vibrations in inertial space, loses this property as soon as the trajectory ceases to be flat. If the pendulum end circumscribes an elliptic trajectory instead of a straight line segment, then this ellipse precesses in the same direction as the material point circumscribes the ellipse itself. In this case, the angular velocity of the ellipse precession is proportional to its area and can be explained by the nonlinearity of the equations of vibrations of a mathematical pendulum [1].A similar phenomenon takes place in an elastic inextensible ring, which is a representative of the “generalized Foucault pendulum” family [1]. If a standing wave is excited in an immovable ring, then this wave is immovable with respect to the ring only in the case of zero quadrature, but if the quadrature is nonzero, then the standing wave precesses with respect to the ring with a velocity proportional to the quadrature value.As in the case of the classical pendulum, this phenomenon can be explained by the nonlinearity of the ring regarded as an oscillatory system.In the present paper, we obtain an explicit formula for calculating the angular velocity of such a precession.

Spectral Properties of Linear Gyroscopic Systems

We study how the natural frequencies of a linear gyroscopic system behave under variations in the kinetic and potential energy matrices as well as in the gyroscopic force matrix.

Analysis of the structure of generalized forces in the Lagrange equations

Stability of mechanical systems in the sense of Thomson and Tait [1] can be judged from the type of forces applied to them. The forces are usually divided into potential (conservative), circular, dissipative, accelerating, gyroscopic, etc. The decomposition itself of generalized positional forces into conservative and properly nonconservative forces is well known for the case in which these forces linearly depend on the generalized coordinates (e.g., see [1–5]). Such a decomposition is associated with the unique representation of an arbitrary matrix of these forces as the sum of symmetric and skew-symmetric parts. Generalized forces linearly depending on the velocities can in a similar way be divided into dissipative and gyroscopic parts. In the present paper, we show how the same decomposition can be performed in the general nonlinear case.

A new shimmy model

Shimmy is a phenomenon of intense angular self-excited vibrations of the wheels of a carriage. Such self-excited vibrations present a serious threat to traffic safety, which accounts for the great interest of researchers in this phenomenon. The problem is of highest importance for the front wheels of aircraft. Usually, the deformation of pneumatic tires is considered to be the main factor responsible for shimmy. Without challenging this thesis, we nevertheless note that this is not the only factor. The shimmy phenomenon can be observed in everyday life in the case of various carriages that often have nothing to do with pneumatics if the wheels are rigid. Below we will show that the theory of polycomponent dry friction fully explains the shimmy phenomenon for absolutely rigid wheels and, hence, is at least one of the factors responsible for shimmy in the general case. The reason why researchers have not taken dry friction into account when explaining shimmy is that the theory of this kind of friction has not been well developed; at the same time one has failed to explain shimmy in the framework of other existing theories.

A new model of shimmy

Shimmy is the phenomenon of intensive angular self-excited vibrations of a vehicle wheel. Such self-excited vibrations seriously threaten the safety ofmotion, which explains scientists’ profound interest in this phenomenon. This problem is most serious for the front wheels of aircraft. Tyre deformation is usually viewed to be the main cause of shimmy. Without casting doubt on this point, we still note that this is not the only cause. The shimmy phenomenon can also be observed in everyday life and for various hand carts, where any reference to tyre elasticity is most often irrelevant if the wheels are rigid.We show that the polycomponent dry friction theory can completely explain the shimmy phenomenon for absolutely rigidwheels and hence polycomponent dry friction is at least one of its causes in the general case.Dry friction has been neglected by scientists when explaining shimmy because dry friction theory has not been sufficiently well developed until recently; at the same time, shimmy cannot be explained in the framework of earlier conceptions.