pendulum kapitza
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pendulum kapitTRANSCRIPT
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1. Pendulum on a Vibrating Support
Consider a pendulum with a vibrating support (Figure 1). The equation of motion is:
(1)
where is the swing angle, is nondimensional time, is the
linear natural frequency for small oscillations near , , the damping ratio (actual to critical), q the relative amplitude of the vibrating support,
and the nondimensional frequency of this excitation.
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Figure 1. Pendulum with an vibrating support
Splitting Motions into Slow and Fast Components
We consider the case where the support is vibrating rapidly at a small amplitude,
i.e., and . Then one can split the total pendulum motion into slow and fast components as follows:
(2)
where z describes the slow motions at the time-scale of free pendulum oscillations, and describes a small overlay of fast motions at the much faster rate of the support vibrations (for memorizing: z: zlow; : phast). We perceive as the slow time scale and as a fast time scale. We are mainly interested in the slow motions z, whereas the details of the fast overlay interests us only by their effect on z.
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Considering (2) as a transform of variables from to z and , we need to specify a constraint to make the transform unique. For this we require the fast-time average of the fast motions should be zero:
(3)
where defines time averaging over one period of the fast excitation with the slow time considered fixed.
To determine the fast motions we substitute (2) into (1), Taylor-
expand for , and obtain:
(4)
where , , and denote small terms of the
order and less (note that and q << 1, so we assume ). To first order the solution of (4) is:
(5)
To determine the slow motions z we average (4) and obtain, using (3):
(6)
or, substituting (5) for , noting that , and dropping small terms of
the order :
(7)
Now, z denotes the slow component of the pendulum angle . To obtain all the details of the pendulum motion we must add to z the small overlay of fast
motions (cf. (2)), but for now we focus on the large, overall component z of the motion. As appears from (7) this z-component is governed by a differential equation similar to the original equation (1), though with the time-dependent
forcing term being replaced by the term . This term describes the average effect of the fast forcing. Note that (7) is autonomous, and thus considerably simpler to solve than is the original equation of motion (1).
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What happens to equilibriums when there is fast excitation?
The static equilibriums for z is and , as appears when letting in (7) and solving for z. So the pendulum is in equilibrium when pointing straight up or down, just as when there is no support excitation. Though, when the
excitation is sufficiently strong, , there is an additional pair of
equilibriums given by . One can show that these are always unstable.
How does the pendulum behave near the down-pointing equilibrium?
We will find out by linearising (7) near :
(8)
where is the effective natural frequency for free oscillations near :
(9)
Thus, free pendulum oscillations near the down-pointing equilibrium occur at a higher frequency when the support is vibrating up and down. For example,
mounting a pendulum clock at a table that vibrates at intensity , the clock
will run faster than if . Also, since the term in (8) is a linear stiffness term, one may talk about the stiffening effect of fast excitation.
...and near the up-pointing equilibrium?
Here we consider the linearisation of (7) about :
(10)
where is the effective natural frequency for free oscillations near :
(11)
Note here that the squared natural frequency changes sign when .
When the sign of is negative, which implies that the inverted pendulum position is unstable. However, when the vibrations of the support
are sufficiently strong, , then turns positive and the inverted pendulum position becomes stable (this result agrees with [4], using quite different
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methods). At such conditions of strong support vibrations, it appears from (10), any slight disturbance of the inverted pendulum will cause it to return to that
position by performing damped oscillations at frequency . But a large disturbance may cause the pendulum to cross the above-mentioned unstable
equilibriums at , and reach the lower stable equilibrium at .
So, nontrivial effects may appear when subjecting a pendulum or any similar structure to fast vibrations. The effects mentioned above are not just mathematical artifacts; they can quite easily be demonstrated in the laboratory. Let's proceed with another example.