is given by € u ¼ _
h
2 u À
2
3 cos h. The first term is of the centrifugal type, forcing the
disk towards the tip of the rod if _
h 6 ¼ 0. The second term represents the effect of
gravity, forcing the disk towards the rod-tip when the rod points downwards
(cosh < 0) or towards the hinge when the rod points upwards (cosh > 0).
With the rod at rest ( _
h 6 ¼ 0) the disk will move at constant acceleration
2
3 cosh
towards u = 0 or u = 1. With the rod oscillating around h = p (pointing downwards) the disk falls off the rod, since the centrifugal term and the gravity
co-operate in driving the disk against the rod-tip. However, with the rod oscillating
around h = 0 (pointing upwards) the two terms counteract, and may possibly balance each other out, on the average. If this equilibrium state is stable, the disk will
be at rest or perform small oscillations somewhere along the rod.
But how should the rod be brought to oscillate in the upright position, against
gravity? To see this we consider (4.79), governing rotations h(s) of the rod. For a
fixed value of u(s) (i.e., the disk is fixed to the rod) the Coriolis term 2au _
u _
h drops
out, and when further sinh % h for small h the equation becomes that of an ordinary
linear pendulum, subjected to parametric excitation. It is a Mathieu equation, for
which it is known that the equilibrium h = 0 can be stabilized for certain conditions
of (small) amplitude q and (high) frequency X (cf. App. C; Nayfeh and Mook 1979;
Panovko and Gubanova 1965). On the other hand, when h 0 the centrifugal
forces cannot hold the disk in position. So, we are seeking conditions under which
the rod can be stabilized in performing small but rapid oscillations near h = 0.
4.6.5 Seeking Quasi-statical Equilibriums
by Averaging
We consider here the existence and stability of the following configuration of the
system: the rod performs small amplitude vibrations near h = 0, and the disk performs small amplitude vibrations near some fixed value of u 2 ]0; 1[. We shall fail
in this respect, as did Blekhman and Malakhova (1986) using quite other methods,
simply because there are no stable equilibriums. Still, the analysis will be illustrative of the phenomenon of vibration induced sliding.
Assuming h % 0 implies sinh % h and cosh % 1. Assuming further that
movements of the disk are much slower than those of the rod, the Coriolis term
2au _
u _
h of (4.79) can be neglected for a first approximation. Since the rod equation
(4.79) is parametrically excited we assume vibrations of the rod to occur at half the
excitation frequency. Thus, we employ to (4.79) a Van der Pol transformation (h, _
h)
! (a sinw,
1
2 aX cos wÞ with a = a(s), w = w(s) =
1
2 Xs + u(s) and _
a sinw + a _
u
cosw 0. Averaging out the rapid variations in w one arrives at the following
equations, which govern slow modulations of the rod amplitude a and phase u:
4.6 Pendulum with a Sliding Disk
241
h
2 u À
2
3 cos h. The first term is of the centrifugal type, forcing the
disk towards the tip of the rod if _
h 6 ¼ 0. The second term represents the effect of
gravity, forcing the disk towards the rod-tip when the rod points downwards
(cosh < 0) or towards the hinge when the rod points upwards (cosh > 0).
With the rod at rest ( _
h 6 ¼ 0) the disk will move at constant acceleration
2
3 cosh
towards u = 0 or u = 1. With the rod oscillating around h = p (pointing downwards) the disk falls off the rod, since the centrifugal term and the gravity
co-operate in driving the disk against the rod-tip. However, with the rod oscillating
around h = 0 (pointing upwards) the two terms counteract, and may possibly balance each other out, on the average. If this equilibrium state is stable, the disk will
be at rest or perform small oscillations somewhere along the rod.
But how should the rod be brought to oscillate in the upright position, against
gravity? To see this we consider (4.79), governing rotations h(s) of the rod. For a
fixed value of u(s) (i.e., the disk is fixed to the rod) the Coriolis term 2au _
u _
h drops
out, and when further sinh % h for small h the equation becomes that of an ordinary
linear pendulum, subjected to parametric excitation. It is a Mathieu equation, for
which it is known that the equilibrium h = 0 can be stabilized for certain conditions
of (small) amplitude q and (high) frequency X (cf. App. C; Nayfeh and Mook 1979;
Panovko and Gubanova 1965). On the other hand, when h 0 the centrifugal
forces cannot hold the disk in position. So, we are seeking conditions under which
the rod can be stabilized in performing small but rapid oscillations near h = 0.
4.6.5 Seeking Quasi-statical Equilibriums
by Averaging
We consider here the existence and stability of the following configuration of the
system: the rod performs small amplitude vibrations near h = 0, and the disk performs small amplitude vibrations near some fixed value of u 2 ]0; 1[. We shall fail
in this respect, as did Blekhman and Malakhova (1986) using quite other methods,
simply because there are no stable equilibriums. Still, the analysis will be illustrative of the phenomenon of vibration induced sliding.
Assuming h % 0 implies sinh % h and cosh % 1. Assuming further that
movements of the disk are much slower than those of the rod, the Coriolis term
2au _
u _
h of (4.79) can be neglected for a first approximation. Since the rod equation
(4.79) is parametrically excited we assume vibrations of the rod to occur at half the
excitation frequency. Thus, we employ to (4.79) a Van der Pol transformation (h, _
h)
! (a sinw,
1
2 aX cos wÞ with a = a(s), w = w(s) =
1
2 Xs + u(s) and _
a sinw + a _
u
cosw 0. Averaging out the rapid variations in w one arrives at the following
equations, which govern slow modulations of the rod amplitude a and phase u:
4.6 Pendulum with a Sliding Disk
241