Page 228 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.3 Lagrange’s Equation 215

7.3 LAGRANGE’S EQUATION
In our previous discussions, we were able to formulate the equations of motion by
three different methods. Newton’s vector method offered a simple approach for
systems of a few degrees of freedom. The necessity for the consideration of forces
of constraints and free-body diagrams in this method led to algebraic difficulties
for systems of higher degrees of freedom.
The energy method overcame the difficulties of the vector method. However,
the energy principle in terms of physical coordinates provided only one equation,
which limited its use to single-DOF systems.
The virtual work method overcame the limitations of both earlier methods
and proved to be a powerful tool for systems of higher DOF. However, it is not
entirely a scalar procedure in that vector considerations of forces are necessary in
determining the virtual work.
Lagrange’s formulation is an entirely scalar procedure, starting from the
scalar quantities of kinetic energy, potential energy, and work expressed in terms
of generalized coordinates. It is presented here as
A. d T (7.3-1)
dt dq^ dq, ^ dq, ~
The left side of this equation, when summed for all the q¿, is a statement of the
principle of conservation of energy and is equivalent to
d { T ^ U ) = 0 (7.3-2)
The right side of Lagrange’s equation results from dividing the work term in
the dynamical relationship dT = dW into the work done by the potential and
nonpotential forces as follows:
dT = dW^^ +
The work of the potential forces was shown earlier to be equal to dW^ =
—dU,'^ which is included in the left side of Lagrange’s equation. The nonpotential

work is equal to the work done by the nonpotential forces in a virtual displacement
expressed in terms of the generalized coordinates. Thus, Lagrange’s equation, Eq.
(7.3-1), is the Qj component of the energy equation
d(T+U)=dW „„ (7.3-3)
We can write the right side of this equation as
= e , + 02 5^2 + • • • (7.3-4)
The quantity is called the generalized force. In spite of its name, can have
units other than that of a force; i.e., if is an angle, has the units of a
moment. The only requirement is that the product 8q^ be in the units of work.

dW = F-dr = F-rdt = mr ■rdt

\mr r, dT = m'r ■rdt = dW