Page 230 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.3 Lagrange’s Equation 217

These three equations can now be assembled into matrix form:
mi 0 0 “ i^V + k2) ~^2 0
0 m2 0 I -h -k2 (^ 2 + ^ 3) -^3
0 0 0 -k . ^3
We note from this example that the mass matrix resul
(d/dt){dT/dq-) - dT/dq- and the stiffness matrix is obtained from dU/dq^.
Example 7.3-2
Using Lagrange’s method, set up the equations of motion for the system shown in Fig.
7.3-2.

Figure 7.3-2.
Solution: The kinetic and potential energies are
T = \mq^ + \jqj
U = jkqj + {k(rq2 - <?,)"
and from the work done by the external moment, the generalized force is
8 W = m { t ) 8 q , .-. 02 =" ^ ( 0
Substituting into Lagrange’s equation, the equations of motion are
mq^ + 2kq^ — krq2 = 0
Jq2 — krq^ kr^q2 = 2)?(r)
which can be rewritten as
m 0 2k -kr
°
0 J Ur -kr kr^ ' " ' i - i |W(<)
_ « ! (
Example 7.3-3
Figure 7.3-3 shows a simplified model of a two-story building whose foundation is
subject to translation and rotation. Determine T and U and the equations of motion.
Solution: We choose u and 6 for the translation and rotation of the foundation and y for
the elastic displacement of the floors. The equations for T and U become
, 2

+ + 2h0 + 3^2) ^ 2^^
u = ik()ir + ~ y\Y
where ii, 0, y,, and y2 are the generalized coordinates. Substituting into Lagrange’s