Stationary interaction between viscoelastic beam and rigid barrier (Q2755249)

This article deals with the investigation of interaction between viscoelastic beam and elastic obstacle with rigidity \(c\). The equation of longitudial vibrations of viscoelastic beam has the form \({1\over c_1^2}{\partial^2u\over\partial t^2}- {\gamma\over \omega}{\partial^3 u\over\partial t\partial x^2}- {\partial^2u\over\partial x^2}=0\), where \(c_1\) is the velocity of longitudinal waves, \(\gamma\) is the absorption coefficient, and \(\omega\) is the circular frequency of excitation. The boundary conditions have the form \(u|_{x=0}= u_0\cos(\omega t-\phi)\), \(\sigma|_{x=l}={c\over S}(\Delta-u|_{x=l}), 0<t<t_1\), \(\sigma|_{x=l}=0, t_1<t<2\pi/\omega\), where \(u_0\) and \(\phi\) are the amplitude and phase shift of excitation; \(\sigma=E\left({\partial u\over\partial x} +{\gamma\over\omega}{\partial^2 u\over\partial t\partial x}\right)\); \(\Delta\) is the distance between the beam end and the obstacle; \(t_1\) is a termination moment of contact; \(S\) is the area of beam cross-section; \(E\) is the Young modulus. By reduction of the considered problem to the Fredholm integral equation of the second kind, the author proves existence of a periodic solution. The phenomenon of double contact of the beam with the obstacle during the period is discovered.