Beam vibrations with time-dependent boundary conditions

A procedure is outlined for the solution of the vibration problem of a Bernoulli-Euler beam with time-dependent boundary conditions. The solution is greatly simplified if the dependent variable in the original partial differential equation can be changed to produce homogeneous boundary conditions and at the same time maintain a homogeneous differential equation. A method for making such a change is given and illustrated by solving a cantilever beam problem with a time-dependent tip displacement.