Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.

A simple nonlinear element model2017In: Acoustical Physics, ISSN 1063-7710, E-ISSN 1562-6865, Vol. 63, no 3, p. 270-274Article in journal (Refereed)

Abstract [en]

We study experimentally the behavior of a nonlinear element, a light plate pressed to the opening in the cavity of an acoustic resonator. Measurements of field oscillations inside and outside the cavity have shown that for large amplitudes, they become essentially anharmonic. The time dependences of displacement of the plate with increasing amplitude of the exciting voltage demonstrates a gradual change in the shape of vibrations from harmonic to half-period oscillation. A constant component appears in the cavity: rarefaction or outflow of the medium through the orifice. We construct a theory for nonlinear oscillations of a plate taking into account its different elastic reactions to compression and rarefaction with allowance for monopole radiation by the small-wave-size plate or radiation of a plane wave by the plate. We calculate the amplitudes of the harmonics and solve the problem of low-frequency stationary noise acting on the plate. We obtain expressions for the correlation function and mean power at the output given a normal random process at the input.

A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.

Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.

Modular solutions2016In: Doklady Mathematics, ISSN 1064-5624, Vol. 94, no 3, p. 708-711Article in journal (Refereed)

Abstract [en]

Solutions to a partial differential equation of the third order containing the modular nonlinearity are studied. The model describes, in particular, elastic waves in media with weak high-frequency dispersion and with different response to tensile and compressive stresses. This equation is linear for solutions preserving their sign. Nonlinear phenomena only manifest themselves to alternating solutions. Stationary solutions in the form of solitary waves or solitons are found. It is shown how the linear periodic wave becomes nonlinear after exceeding a certain critical value of the amplitude, and how it transforms into a soliton with further increase in the amplitude.