Appendix: All Solutions to the Problems
925
The Lagrange function is
L = T − V − U =
1
2
m 1 y 1
2
+
1
2
m 2 y 2
2
+ m 1 gy 1 + m 2 gy 2
−
1
2
k 1 y
2
1 −
1
2
k 2 (y 2 − y 1 )
2
The functional established by the Hamilton’s principle is
J [y 1 , y 2 ] =
t 1
t 0
1
2
m 1 y
2
1 +
1
2
m 2 y
2
2 + m 1 gy 1 + m 2 gy 2
−
1
2
k 1 y
2
1 −
1
2
k 2 (y 2 − y 1 )
2
dt
The Euler equations of the functional namely the differential equations of motion
are
m 1 y 1
+ k 1 y 1 − k 2 (y 2 − y 1 ) − gm 1 = 0
m 2 y 2
+ k 2 (y 2 − y 1 ) − gm 2 = 0
9.12 As shown in problem 9.12 figure, a homogeneous circular cylinder that the
radius is r , the mass is m makes pure rolling within the cylindrical groove of radius
R, using the Hamilton’s principle to derive the equation of small vibration and period
of vibration of the circular cylinder.
Solution The system has only one degree of freedom, choosing θ as the generalized
coordinate. The rotational inertia of the circular cylinder is J =
1
2
mr
2 .
The velocity of the center of the circular cylinder is
v = (R − r ) ˙
θ
The angular velocity of the circular cylinder is
ω =
R − r
r
˙
θ
The kinetic energy of the system is
T =
1
2
mv 2 +
1
2
J ω 2 =
1
2
m[(R − r ) ˙
θ] 2 +
1
2
×
1
2
mr 2
(R − r ) ˙
θ
r
2
=
3
4
m[(R − r ) ˙
θ] 2
The potential energy of the system is
V = −mg(R − r ) cos θ