To solve problems involving the forced response of strings using Lagrange’s equation, we also need the kinetic energy. The kinetic energy for a differential length of string is
Recalling that the longitudinal displacement u was shown previously to be less significant than the transverse displacement v and to uncouple from it for small – perturbation motions about the static-equilibrium state, we may now express the
kinetic energy of the whole string over length I as
2.2.2 Virtual Work of Applied, Distributed Force
To solve problems involving the forced response of strings using Lagrange’s equation, we also need a general expression for the virtual work of all forces not accounted for in the potential energy. These applied forces and moments are identified most commonly as externally applied loads, which may or may not be a function of the response. They also include any dissipative loads, such as those from dampers. To determine the contribution of distributed transverse loads, denoted by f (x, t), the virtual work may be computed as the work done by applied forces through a virtual displacement, viz.
where the virtual displacement Sv also may be thought of as the Lagrangean variation of the displacement field. Such a variation may be thought of as an increment of the displacement field that satisfies all geometric constraints.