We consider a gyroscopic system under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating bodies of revolution being in frictional contact. The spectrum of the unperturbed gyroscopic system forms a "spectral mesh" in the plane "frequency -gyroscopic parameter" with double semi-simple purely imaginary eigenvalues at zero value of the gyroscopic parameter. It is shown that dissipative forces lead to the splitting of the semi-simple eigenvalue with the creation of the so-called "bubble of instability" - a ring in the three-dimensional space of the gyroscopic parameter and real and imaginary parts of eigenvalues, which corresponds to complex eigenvalues. In case of full dissipation with a positive-definite damping matrix the eigenvalues of the ring have negative real parts making the bubble a latent source of instability because it can "emerge" to the region of eigenvalues with positive real parts due to action of both indefinite damping and non-conservative positional forces. In the paper, the instability mechanism is analytically described with the use of the perturbation theory of multiple eigenvalues. As an example stability of a rotating circular string constrained by a stationary load system is studied in detail. The theory developed seems to give a first clear explanation of the mechanism of self-excited vibrations in the rotating structures in frictional contact, that is responsible for such well-known phenomena of acoustics of friction as the squealing disc brake and the singing wine glass.
An axially symmetric shell, like a wine glass can easily produce sound when a wet finger is rubbed around its rim or wall, as it was observed already in 1638 by Galileo Galilei in his Dialogues Concerning Two New Sciences [1,8,34]. This principle is used in playing the glass harmonica invented by Benjamin Franklin in 1757, which he called "armonica", where, in order to produce sound, one should touch by a moist finger an edge of a glass bowl rotating around its axis of symmetry [15,22,34,53]. This remarkable phenomenon has been studied experimentally; see e.g. [40]. However, an adequate analytical theory for its description seems to be still missing. Another closely related example of acoustics of friction is the squealing disc brake [22,26].
This mechanical system produces sound due to transverse vibrations of a rotating annular plate caused by its interaction with the brake pads. Despite intensive experimental and theoretical study, the problem of predicting and controlling the squeal remains an important issue [4,12,19,22,24,26,34,47,48,51]. Significant but still poorly understood phenomena are squealing and barring of calender rolls in paper mills causing an intensive noise and reducing the quality of the paper [35].
The presence of multiple eigenvalues in the spectra of free vibrations of axially symmetric shells and plates is well-known. Already Rayleigh, studying the acoustics of bells, recognized that, if the symmetry of a bell were complete, the nodal meridians of a transverse vibration mode would have no fixed position but would travel freely around the bell, as do those in a wine glass driven by the moistened finger [22]. This is a reflection of the fact that spectra of free vibrations of a bell, a wine glass, an annular plate and other bodies of revolution contain double purely imaginary semi-simple eigenvalues with two linearly independent eigenvectors. Rotation causes the double eigenvalues of an axially symmetric structure to split [2]. The newborn pair of simple eigenvalues corresponds to the forward and backward travelling waves, which propagate along the circumferential direction [2,3,4,9,16,19,20]. Viewed from the rotational frame, the frequency of the forward travelling wave appears to decrease and that of the backward travelling wave appears to increase, as the spin increases. Due to this fact, double eigenvalues originate again at non-zero angular velocities, forming the nodes of the spectral mesh in the plane ’eigenfrequency’ versus ‘angular velocity’. The spectral meshes are characteristic for example for the rotating circular strings, rings, discs, and cylindrical and hemispherical shells.
The phenomenon is apparent also in hydrodynamics, in the problem of stability of a vortex tube [27] and in magnetohydrodynamics (MHD) in the problem of instability of the spherically symmetric MHD α 2 -dynamo [43].
It is known that striking the wine glass excites a number of modes, but rubbing the rim with a finger or bowing it radially with a violin bow generally excites a single mode [15]. The same is true for the squealing disc brake [26,44,45,46,48,51]. For this reason, we formulate the main problem of acoustics of friction of rotating elastic bodies of revolution as the description of the mechanism of activating a particular mode of the continuum by its contact with an external body.
In case of the disc brake, the frictional contact of the brake pads with the rotating disc introduces dissipative and non-conservative positional forces into the system [48,51]. Since the nodes of the spectral mesh correspond to the double eigenvalues, they are most sensitive to perturbations, especially to those breaking the symmetries of the system. Consequently, the instability will most likely occur at the angular velocities close to that of the nodes of the spectral mesh and the unstable modes of the perturbed system will have the frequencies close to Figure 1: The spectral mesh of system (1) when δ = ν = κ = 0. that of the double eigenvalues at the nodes. This picture qualitatively agrees with the existing experimental data [15,22,26,44,45,46].
In the following with the use of the perturbation theory of multiple eigenvalues [33,38,39,43] we will show that independently on the definiteness of the damping matrix, there exist combinations of dissipative and non-conservative positional forces causing the flutter instability in the vicinity of the nodes of the spectral mesh for the angular velocities from the subcritical range. We will show that zero and negative eigenvalues in the spectrum of the damping matrix encourage the development of the localized subcritical flutter instability while zero eigenvalues in the matrix of non-conservative positional forces suppress it. Explicit expressions describing the movements of eigenvalues due to change of the system parameters will be obtained. Conditions will be derived for the eigenvalues to move to the right part of the complex plane. Approximations of the dom
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