## Abstract

A golf ball having special dimples flies better than an analogous smoothed one. A surprise is that there is a range of Reynolds numbers for which the turbulent drag is somewhat less than that in the laminar case. Analogies always meet together and are accomplished themselves in the physics. We have traced an effect similar to the above mentioned in the area of solid mechanics when the nonlinear system passes through a sequence of bifurcations. In mechanical engineering, the role of such a system can play a solid-state wave gyro entering the family of MEMS/NEMS. It is known that a circular Foucault pendulum can serve as an angular sensor. Standing waves in a thin-walled elastic axisymmetric resonator of a solid-state wave gyro, mounted on a rotating platform, can also detect a rotation rate. Because there are no typical mechanical parts there, such wave sensors have advantages for long-term space missions. However, to maintain the functionality and sensitivity of a wave gyro in practice, the driving of standing waves requires a sophisticated feedback control. Nonetheless, we have demonstrated that such a gyro can operate without any feedback at the expense of the natural nonlinearity of the resonator in a postbifurcation regime.

### Keywords

- wave solid-state gyro
- Foucault pendulum
- spring pendulum
- bifurcation
- stability

## 1. Introduction

It is well known that the aerodynamic drag is proportional, first of all, to the kinetic energy of a flow. Moreover, the drag depends on a lot of other almost uncontrolled factors such as the shape, size, inclination of the object in the flow, etc. Implicitly, all these are accumulated all together in the following formula for the hydrodynamic resistance:

Physical analogies meet together and are accomplished themselves in most applications. Can we find out an effect similar to the abovementioned dimpled ball that flies better in the area of solid mechanics? We know that the hydrodynamic drag evolves throughout the sequence of bifurcations. But any bifurcation is inherent in nonlinear systems. Therefore, our study should deal with nonlinear mechanical systems. In mechanical engineering, the machines, sensors, and other useful devices are of interest. One of them is a solid-state wave gyro from the family of microelectromechanical system/nanoelectromechanical system (MEMS/NEMS) devices. Let us briefly touch some key points of this topic.

In the nineteenth century, people held no doubt that the Earth is turning about its axis. However, there were no direct proofs. For this reason, Léon Foucault had managed in 1851 his famous experiment with the giant pendulum. The heart of his experiment is the inertia of the pendulum or the resistance to change the motion. This means that the swing plane remains to be fixed in Newton’s absolute frame of references; moreover, this evidence can be traced with the naked eye due to the slow clockwise veering since the Earth rotates anti-clockwise. However, Foucault’s pendulum has one essential flaw, namely the dependence on the latitude and its giant size suppressing the energy dissipation for a while. That is why, in 1852, though an idea, we should note, was not new, Foucault produced a compact and convenient gyroscopic setup based on the immobility of the axis of a rotating mass. It was a prototype of a conventional gyro providing needed horizons at sea and then in the air. The irony of fate, when the twentieth century was ending, the old idea came back to use a Foucault-pendulum-based device as a high precision gyroscope, although, in the other role of a thin axisymmetric solid-state resonator, associated with satellite guidance systems intended for long-term missions extending up to 15 years [6].

In 1890, Bryan demonstrated that the revolving of the standing wave nodes records effectively the rotation of the elastic resonator, as equally as a material point tends to conserve the spatial position in Newton’s space. This evidence was not so new. Yet before Bryan’s experiments, it was well known that the plane of transverse vibration of a straight wire will remain fixed in space instead of turning, though the wire rotates slowly about its axis during oscillations. Bryan noticed that when the vibrating body is such as a bell, rotation about its axis will produce an intermediate effect: the nodal meridians revolve with angular velocity less than that of the resonator that exhibits a new finding which yet nothing is easier than to be verified experimentally. He selected a champagne glass, then struck it to get a pure tone and when the glass turns around, Bryan heard sound beats demonstrating that the nodal meridians do not remain fixed in Newton’s absolute space. He evaluated that the nodal angular velocity is about 3/5 of this almost hemispherical resonator [7]. Although, only in the second half of the twentieth century, Bryan’s effect gets a wide extent turning into a concept of a prototype of an angle sensor that possesses a lot of advantages compared with a conventional gyro because its core is a monolith [8, 9, 10]. Moreover, if the power of the wave gyro is lost for a short time, the resonator conserves the angular rate, so that when the power returns, the gyro need not be reinstalled, etc., unlike, for instance, optical gyroscopes.

Let us now recall some mathematical aspects of the problem along the natural evolution of ideas. The role of the most simple mechanical system, appearing as an abstract oscillatory gyro *in vitro*, can play a circular Foucault pendulum. The corresponding mathematical model can be given by the following two differential equations:

where *in vita*, the pendulum should be excited to vibrate. Moreover, the amplitude of oscillations has to be maintained on some sensitive level because of energy dissipation. The presence of damping requires permanently pumping the energy into these oscillations by external forces for a permanent operating.

Almost a half century later, Bryan has derived his equation describing two similar vibration forms which, traveling toward a thin rotating ring, produce phenomena of beats, and which, in the case of high-frequency oscillations like those of sound, can detect effects of slow rotation about the sensitivity axis:

Here, *in vacuo*, that is, in the absence of damping. However, the energy dissipation always presents in nature. Therefore, this system requires some external feedback control to supply the wave motion. The principal parametric resonance is a good way to excite such a gyro by driving the axisymmetric tension

## 2. Spring pendulum

Before all the theoretical approaches related to a wave resonant solid-state gyro, it may be of place to consider a most simple mechanical system which behaves analogously and contains no excess detail, in order to explain from first principles why the nonlinearity stabilizes unstable parametric oscillations, at the expense of resonance experienced in the mechanical system. The best candidate seems to be a spring pendulum.

Therefore, let us consider a pendulum swinging in a plane and consisting of a bob of mass

Alternatively, we may follow the Hamiltonian approach:

and then writing the following set of Hamiltonian equations:

where

Let us now consider small-but-finite oscillations of the pendulum. In this case, the general solution to the set (6) can be represented as follows:

to look for solutions to Eq. (5) with unknown amplitudes

under the one assumption that the following integer-valued ratio between the frequencies, manifesting the primary parametric resonance, that is,

Moreover, Eq. (8) posses one more additional integral of motion:

where

Let us now refer to the damped forced motion of the pendulum. In this case, equations governing the motion (5) can be easily modified to the following form:

where

These equations take in a natural manner into account the so-called triad-angle locking phenomenon described in [2], when both the phase of external force and the phase of radial oscillations have to be coupled, that is,

In the case of stationary motion, when the amplitudes are constants, Eq. (12) produces the following algebraic set of equations:

having the following solutions:

and

These solutions, Eqs. (14) and (15), are plotted in Figure 1. As we can see, the dimensionless parameter

To examine the stability properties without any excess mathematical preparations, let us analyze the power of drag forces using the energy balance resulted from the set (12):

For the stationary orbits, the left-hand term of this equation is zero. Since the right-hand term, representing changes of the average kinetic energy in time, is also zero, then the power of drag forces should be, in turn, proportional to the kinetic energy with a negative sign. The diagram of the average kinetic energy as the function of control parameter

Let us now refer to the original Eq. (11) to verify results yielded by the modulation theory. To plot the numerical results, we have used the following transform:

Since we are studying a damped forced motion, the initial conditions to the governing equations rewritten in terms of either the physical or the dimensionless variables

In new dimensionless variables

Mathematically, the above results represent direct proofs that in our case the modulation theory has more advantages in comparison with numerical investigations. At small and even moderate oscillations, this asymptotic approach allows us to make up a complete parametric analysis of the system under the study. Pragmatically, stability properties demonstrated using the above example of a spring oscillator in post-bifurcation regime can be used when designing an idea of a high-precision angular sensor which can appear in the form of a circular Foucault pendulum, in most simple case, or as a solid-state wave gyro with axisymmetric resonator in more general case (for instance, see [4, 5] and references therein).

Gradually, one may always construct an appropriate theory by further developing the model given by Eq. (1). Nonetheless, here we decided to invoke more general case in our consideration, namely a thin-ring resonator of which is described in detail in [11].

## 3. Thin-ring resonator

In order to define the in-plane position in a thin-ring resonator, thickness

if the ring rotates uniformly in time

Equations of motion are derived from the theory of thin-walled shells that uses Kirchhoff-Love hypotheses. So, the field of displacements in the ring is expressed as

In the rotating frame of references, the Lagrangian density of the system reads

where

where

where

Finally, we should provide the set (19) by the periodicity conditions

### 3.1. Dispersion relation

For studying the wave propagation in the rotating frame of references, it is convenient to define linear modes of oscillations in the ring on the fixed platform. If the rotation is absent, then the normal modes of vibrations represent standing waves being a superposition of two waves traveling toward identical wave numbers, frequencies as well as amplitudes. First of all, we should understand how the spectrum of these oscillatory modes would change in the uniformly rotating ring. Are there the standing modes taking place? And if these cannot be detected, then, what thing should we invoke instead into our study? We know that the precession of waves appears as a reaction on the rotation. Therefore, there will be expected some asymmetry in the polarization vectors of traveling waves. One more finding is we know that the precession rate of waves has to be proportional to the difference between the frequencies in the wave pair as if those compose a standing wave in the ring being at rest. Let the angular rate in the ring be constant and then the wave precession should appear as some sort of kinematic reactions on the rotation of the ring platform. But if the platform rotates with acceleration, then we can expect a kind of some dynamical response that may appear as essentially differ from kinematic one. Fortunately, experiments with revolving axisymmetric bodies tell us that the expression for the uniformly rotating ring, which will be done subsequently, is equally valid in both cases of the uniformly rotating ring and in the motion with arbitrary, but moderate, angular acceleration [8].

In the case of uniform rotating, a simple solution of the linearized set (19) is given by

Here, we can trace the appearance of the constant extension in radial direction caused by the centrifuge force. Notice that the amplitudes,

The high- and low-frequency branches are indexed by

Let the angular velocity be zero and then the dispersion relation turns into the simple algebraic equation

Two identical ones of all the four roots of Eq. (23) are shown in Figure 7. The low-frequency branch refers in general to the bending modes while the high-frequency ones reply mainly to the circumferential modes of extension.

In the case of a small angular rate, that is,

It may not, perhaps, be out of place to note that these roots possess by asymmetry because of the angular rate. Nonetheless, each frequency,

## 4. Nonlinear resonant coupling between the axisymmetric radial oscillation and two similar bending wave forms being in phase

This section invokes briefly some general points from the common theory of waves, including understanding phenomena of resonance experienced in nonlinear mechanical systems and some mathematical preparations which are necessary to investigate the given problem from the viewpoint of the perturbation analysis, using methods of slowly varying amplitudes and phases. Then, we generalize the problem by considering the case of damped forced oscillations in a thin circular ring. This generalization of the problem leads to a thought on how to excite stable wave precession regimes which need no feedback to repair unwanted motions in the solid-state wave gyro.

### 4.1. Triple-mode resonant coupling between waves in the ring

Truncated equations or evolution equations describing the modulation phenomena of amplitudes and phases can be directly obtained using the following anzats [12, 13, 14, 15]:

which represent an approximated solution to Eq. (19). Here,

We assume that the angular rate and acceleration of the ring are small enough, that is,

where *resonant triplet* while the synchronicity (26) is associated with the so-called *phase-matching conditions*.

One more convenient way of obtaining the evolution equation is walking down along the Hamiltonian formalism. Let us consider the average Lagrangian of the system

in order than to expand this one in a formal series in small parameter

Here, zero-order approximation term, that is,

In this case of resonant coupling between the axisymmetric radial oscillation and two low-frequency bending waves, being in phase (26), this average Hamiltonian is rearranged in more pragmatic form

Recall that the bending wave frequency *principal parametric resonance*. His Hamiltonian produces a set of following evolution equations:

where

Now, we use the following transform of variables:

that allows us to rewrite the set (29) by getting rid of Coriolis terms:

Eq. (32) have now obtained a standard form similar to Euler’s kinematic equations describing rotation of a rigid body with a fixed point. This set can be integrated exactly in terms of the Jacobi elliptic functions [17]. Also, here is a place to note that in the theory of nonlinear waves similar equations describe *break-up instability* phenomena when the high-frequency mode becomes unstable triad with respect to small low-frequency perturbations [12].

### 4.2. Resonant excitation of the gyro

Let us invite in our consideration the generalized forces describing the damping and forcing of the ring resonator over the axisymmetric form of oscillation. By substituting the terms

that can be rewritten, after one more exchange of variables;

where

while the second would be

Here, we have introduced a new notation:

Now, we may return to our preliminary study over the spring pendulum to be acquainted in similarities and analogies. Indeed, the first stationary solution (34) appears as stable only within the range

When returning to the old notation, the solution of our problem can be expressed as it follows:

where

is the precession rate. Typical patterns of the wave precession are shown in Figure 8.

## 5. Conclusion

Léon Foucault has made a sensation with his famous giant pendulum experiment in the Panthéon in Paris in 1851. That time, people could perceive the inertia of the pendulum with his naked eye to have the first direct proof that the Earth is turning anti-clockwise. Moreover, this experiment has demonstrated that a pendulum could be used as a vibratory gyro. However, a flaw of the Foucault pendulum is the dependence on the latitude. A device was needed that was unaffected by the latitude. In 1852, Foucault proposed an angular sensor based on the immobility of the axis of a rotating mass. Unlike the pendulum, this sensor could detect a fixed direction in Newton’s absolute space. Recall that a torque directing the axis of rotation of the rotor of the gyro is known as the precession. For the precision, however, this gyro required an exquisite construction, since the gyro’s rotor should be balanced as well as possible.

In 1890, Bryan investigated the nature of the beats which may be heard when a vibrating cylinder or other axisymmetric thin-walled shell of revolution has been involved in a rotatory motion about its axis [7]. Perhaps, this study had been inspired as a reply to A. E. H. Love, famous for his work on the mathematical theory of elasticity. In the paper [20], Love supposed that unless a body was revolving with the angular velocity comparable with the frequencies of the vibrations, the latter would not be almost affected. The only important effect of rotation would be an extension because of the centrifugal force. Bryan has corrected that in the axisymmetric shells at high-frequency vibrations phenomena of beats may be observed, which appears as the most noticeable effects of the rotation. It is probable that Bryan had no idea how to utilize his finding. Though those times people were in controversy on how far theory of thin-walled shells elaborated by Lord Rayleigh is capable of practice, is this perhaps just a sort of one more abstract theory? Lo and behold, initially conceived in 1890 through the observation of beats from a ringing wine glass, the concept was lost until uncovered in 1965. Due to technologies of the second half of the twentieth century, perhaps in times of Vietnam War, this physical effect has inspired a concept of a solid-state wave gyro [6]. Because there are no typical mechanical parts, these wave sensors have a lot of advantages for long-term space missions. However, to maintain the functionality as well as the sensitivity of a conventional wave gyro in practice, the driving of standing waves requires somewhat sophisticated feedback control. This chapter demonstrates that when both the primary resonant pumping over the axisymmetric mode of oscillations and advantages of the principal parametric resonance are combined, such a gyro can operate without any feedback at the expense of the natural nonlinearity of the resonator in a post-bifurcation regime.