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.

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).

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

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.