Vostrukhov A.V., Metrikine A.V.


The work of a source maintaining a uniform motion of a load along an elastically supported membrane


Acoustical Physics 46(2), 139-143, 2000


Summary. The steady state response of a membrane on an elastic foundation to a uniformly moving constant point load is investigated. Analytic expressions for the vertical displacement of a membrane are derived. Two qualitatively different cases are considered: sub-critical case, when the load velocity  is smaller than the velocity of waves in the membrane , and supercritical case, when . A general expression for the work per unit time provided by an external source maintaining the uniform motion of the load along the membrane is obtained. It is shown that neglecting energy losses in the membrane, this work is equal to zero for the sub-critical case and tends to infinity for supercritical motion. The infinite power needed to maintain the load motion is a consequence of discontinuity of the membrane displacement, which takes place at the borders of the ĞMach angleğ in the supercritical case. This discontinuity as it is shown in the paper can be avoided if one takes into account an internal friction in the membrane. Solution of the problem with the internal friction is found in the form of a single integral and then analyzed numerically. On the base of this analysis dependency of the work by the external source is studied as a function of the load velocity.