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

Analytical expressions for the transverse displacement of a membrane mounted on an elastic base and subjected to a uniformly moving load are derived. Two cases are considered: a subcritical motion when the load velocity V is less than the wave propagation velocity in the membrane c and a supercritical motion when V > c. A general expression for the work provided by a source maintaining the uniform motion of a concentrated load along the membrane is obtained. In the absence of the energy loss in the membrane, this work is equal to zero for the subcritical motion and tends to infinity for the supercritical motion. The latter result is a consequence of the discontinuity of the solution obtained for the membrane displacement. With the introduction of the internal friction in the membrane, it is possible to eliminate this discontinuity. On the basis of the numerical solution to the problem with internal friction, the dependence of the work of the external source on the load velocity is analyzed.