Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

Active damping of the forced vibration of a hinged beam with piezoelectric layers,geometrical and physical nonlinearities taken into account

The problem of forced vibration of a hinged beam with piezoelectric layers is solved. Issues of mechanical and electric excitation of vibration and the possibility of damping mechanically induced vibration by applying a voltage to the electrodes of the piezolayers are studied. The effect of the physically nonlinear behavior of the passive layers on the response of the sensor layer and entire structure and the effect of geometric nonlinearity on the behavior of the structure and sensor layer are analyzed. The interaction of physical and geometrical nonlinearities for transient and stationary processes is studied Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 118–136, January 2009.