W

This is an example of a two-degree-of-freedom, conservative system.

A.6 Lagrange's Equations for Nonconservative Systems

If the system is nonconservative, then, in general, there will be some forces (conservative) that are derivable from a potential function, P(q\, <72. • • •) and some forces (non-conservative) that are not. Those forces for which a potential function does not exist must be introduced by first determining their virtual work. The coefficient of the virtual displacement Sq, in the virtual work expression is the generalized force, here denoted by Q, (i — 1, 2,...). In this instance it is convenient to introduce what is called the Lagrangean

L — K — P (A.34) and write the general form of Lagrange's equations as d /3L\ 3L

Example 7

Rework Example 5 with a dashpot of constant c connected in parallel with the spring.

Solution'. The system with a dashpot is nonconservative. Hence, we use Lagrange's equations in form of Eq. (A.35). The kinetic and potential energies are the same as in Example 5. To calculate the Q for the dashpot force, use the definition that Q is the coefficient by which the generalized coordinates must be multiplied to obtain the work done. In any small displacement Sx, the work done by the dashpot force —cx is —ci Sx. Hence, —ci itself is the generalized force associated with the dashpot. The Lagrangean is

Wx2 kx2

0 0