$$\begin{aligned} P &= \boldsymbol{P}' > 0\\ \begin{bmatrix} -r^2 \boldsymbol{P} & \bullet\\ A + B\big{(D\_i \boldsymbol{F} + D\_i^- \boldsymbol{H}) + q\boldsymbol{I} & -\boldsymbol{P}^{-1} \end{bmatrix} &+ \varepsilon \begin{bmatrix} 0\\ M \end{bmatrix} \begin{bmatrix} 0\\ M \end{bmatrix}^\prime + \varepsilon^{-1} \begin{bmatrix} N\\ 0 \end{bmatrix} \begin{bmatrix} N & \boldsymbol{0} \end{bmatrix} +\\ \boldsymbol{\nu} \begin{bmatrix} 0\\ M \end{bmatrix} \begin{bmatrix} 0\\ M \end{bmatrix}^\prime + \boldsymbol{\nu}^{-1} \begin{bmatrix} \left(\boldsymbol{D}\boldsymbol{i} + \overline{\boldsymbol{D}}\boldsymbol{H}\right)^\prime \boldsymbol{N}\_b^\prime\\ \boldsymbol{0} \end{bmatrix} &\sim \begin{bmatrix} \boldsymbol{N}\_b \begin{bmatrix} \boldsymbol{D} \boldsymbol{i} + \overline{\boldsymbol{D}}\boldsymbol{H} \end{bmatrix} \begin{bmatrix} \boldsymbol{N}\_b \end{bmatrix} \begin{bmatrix} 0\\ M \end{bmatrix} \end{aligned} \tag{35}$$

or equivalently,

The last matrix inequality can be linearized by pre- and postmultiplying by [P�1

32 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

Note that the above condition is only a sufficient condition for regional pole placement in one circle. Since we have to achieve pole placement in the area between the two circles, one for ts

The regional pole placement with robust saturated state-feedback control (12) for the uncertain system (8) can be achieved if there exist X=X0 > 0,ε1 > 0, ε2 > 0,ν1 > 0, ν2 > 0,Y and a feasible

AX þ B DiY þ DiZ � � þ q1X �X þ ð Þ ε1 þ ν1 MM0

AX þ B DiY þ DiZ � � þ q2X �X þ ð Þ ε2 þ ν2 MM0

and radius r if and only if there is a feasible solution to the following LMI:

A þ ΔA þ ð Þ B þ ΔB DiF þ D�

1X • ••

NX 0 �ε1I • Nb DiY þ DiHX � � 0 0 �ν1I

2X • ••

NX 0 �ε2I • Nb DiY þ DiHX � � 0 0 �ν2I

> 1=ρ Zj • X � �

From (26), the poles of the closed-loop uncertain system (13) lie inside the circle of center (�q, 0)

" #

�r2P •

i H � � þ qI �P�1

xkþ1 ¼ ð Þ A þ ΔA xk þ ð Þ B þ ΔB :sat uð Þk (29)

• •

• •

> 0 (32)

F ¼ YX�1 (33)

< 0

< 0 (30)

< 0 (31)

(34)

applying fact 1 and substituting P�1 = X, FX = Y.

Consider the uncertain system with saturated control

�r2

�r2

Moreover, the saturated robust pole placer is given by

P ¼ P 0 > 0

and the other for ζ, we have theorem 1.

Design case 2: saturated control

solution to the following LMIs:

for i = 1,…, η; j = 1, … , m.

Proof

Theorem 2

, I, I, I], that is,

$$
\begin{bmatrix}
A + B\left(D\_iF + \overline{D}\_iH\right) + qI & -P^{-1} + (\varepsilon + \nu)MM' & \bullet & \bullet \\
N & 0 & -\varepsilon I & \bullet \\
N\_b\left(D\_iF + \overline{D}\_iH\right) & 0 & 0 & -\nu I
\end{bmatrix} < 0
\tag{36}
$$

By post- and premultiplying (36) by P�1 , that is, by applying the congruence transformation and substituting P�1 = X, one gets (30). This completes the proof.

#### 4. Design of vehicle active suspension control

In this section, we present a design example to illustrate the effectiveness of the proposed controller design method. Unlike the previous approaches, the proposed design introduces a digital computer or a microprocessor into the control loop as given in theorem 2. It is used to design an active suspension system for the quarter-vehicle model shown in Figure 1 [9].

Note that active forces generated by hydraulic actuators and considered as control inputs are bounded because of actuator saturation. The system parameters and nominal values are shown in Table 1.

The normalized input is bounded as |u(t)|≤1, and the active force is bounded by us = allowable spring stroke (�0.08 m) \*spring constant.

Table 1. Quarter-vehicle active suspension parameters.

Due to different passenger load variations, it is assumed that the mass Ms varies between 250 to 390 kg. The system is discretized with the zero order hold method at a sampling time Ts=0.001 sec. Therefore, the discrete time norm-bounded model is obtained as

$$A = \begin{bmatrix} 0.9997 & 0.000986 & 0.002476 & -0.0009852 \\ -0.05546 & 0.9969 & -0.007783 & 0.003106 \\ 0.0002228 & 1.245e-5 & 0.9975 & 0.0009867 \\ 0.4433 & 0.02485 & -4.934 & 0.9727 \end{bmatrix} \tag{37}$$

$$B = \begin{bmatrix} 2.089e-5 & 0.004622 & -1.857e-5 & -0.03694 \end{bmatrix}^\prime \tag{38}$$

With uncertainty matrices

$$M = \begin{bmatrix} -0.0004 & -0.1367 & 0 & -0.0016 \end{bmatrix}^{\prime} \tag{39}$$

$$N = \begin{bmatrix} -0.0729 & -0.0037 & -0.0102 & 0.0041 \end{bmatrix} \tag{40}$$

$$\mathbf{N\_b} = 0.0061\tag{41}$$

The LMIs (30) are solved to get the saturated control gain matrix as

Case 1: step road test

Case 2: road bump test

this case is given by

Figure 6. Step response at light load.

0.005

0.01

0.015

0.02

suspension deflection,m

0.025

0.03

0.035

0.04

0.045

0

As shown in Figure 5, all the closed-loop poles lie inside the desired domain.

damps the oscillations in about 0.3 s without overshoot (Figures 6 and 7).

step response

0 0.5 1

passive suspension

active suspension

time,s

In general, road disturbances can be classified as shock or vibration. For shock disturbances, the proposed controller is tested for two cases: step and bump road. The frequency response is studied when the system is subjected to resonant frequency disturbance forces for the vibration case.

For the simulation, we assume that the vehicle is subject to a 500 N unit step input due to a step road change. Two cases are considered: light and heavy passenger loads. With passive suspension, the vehicle will oscillate for an unacceptable long time, about 3.5 s, with large overshoot (Figures 6 and 7). This might damage the suspension system, whereas the active suspension

In order to study the system response due to road bump, the case of an isolated bump in an otherwise smooth road surface is considered [9]. The corresponding ground displacement in

F ¼ �½ � 402:7340; �47:9208; 546:4541; 16:1682 (42)

Design of Robust Digital Pole Placer for Car Active Suspension with Input Constraint

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35

0 0.5 1

step response

time,s

normalized control

0

For passenger comfort, the oscillations due road bumps should be damped out within 1 s with a minimum damping ratio ζ = 0.25 [20]. To achieve the first objective, we select r1 = 0.9960 and q1 = 0 for the first circle, while for the second, r2 = 0.68 and q2 = �0.31.

Figure 5. Closed-loop poles with system uncertainties using saturated control.

The LMIs (30) are solved to get the saturated control gain matrix as

$$F = [-402.7340, -47.9208, 546.4541, 16.1682] \tag{42}$$

As shown in Figure 5, all the closed-loop poles lie inside the desired domain.

In general, road disturbances can be classified as shock or vibration. For shock disturbances, the proposed controller is tested for two cases: step and bump road. The frequency response is studied when the system is subjected to resonant frequency disturbance forces for the vibration case.

#### Case 1: step road test

Due to different passenger load variations, it is assumed that the mass Ms varies between 250 to 390 kg. The system is discretized with the zero order hold method at a sampling time

B ¼ ½ � 2:089e � 5 0:004622 �1:857e � 5 �0:03694 0

M ¼ �½ � 0:0004 �0:1367 0 �0:0016 0

For passenger comfort, the oscillations due road bumps should be damped out within 1 s with a minimum damping ratio ζ = 0.25 [20]. To achieve the first objective, we select r1 = 0.9960 and

with controller

z-approx

q1 = 0 for the first circle, while for the second, r2 = 0.68 and q2 = �0.31.

z-exact

Figure 5.  Therefore, the discrete time norm-bounded model is obtained as

34 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

A ¼

With uncertainty matrices

0

0.2

0.4

0.6

0.8

1

> For the simulation, we assume that the vehicle is subject to a 500 N unit step input due to a step road change. Two cases are considered: light and heavy passenger loads. With passive suspension, the vehicle will oscillate for an unacceptable long time, about 3.5 s, with large overshoot (Figures 6 and 7). This might damage the suspension system, whereas the active suspension damps the oscillations in about 0.3 s without overshoot (Figures 6 and 7).

#### Case 2: road bump test

In order to study the system response due to road bump, the case of an isolated bump in an otherwise smooth road surface is considered [9]. The corresponding ground displacement in this case is given by

Figure 6. Step response at light load.

Figure 7. Step response at heavy load.

$$\mathbf{x}\_{0} = \begin{cases} \frac{A}{2} \left[ 1 - \cos\left(\frac{2\pi V}{L} t\right) \right], & 0 \le t \le \frac{L}{V} \\\\ 0, & t > \frac{L}{V} \end{cases} \tag{43}$$

well as it does not violate the constraints of suspension stroke (8 cm) and hydraulic actuator force limit of 1.5 KN. Active suspension also provides better passenger comfort than the

normalizedcontrol force

0

0.2

0.4

0.6

0.8

Design of Robust Digital Pole Placer for Car Active Suspension with Input Constraint

0 0.5 1 1.5 2

time,s

normalized control-light load

http://dx.doi.org/10.5772/intechopen.70587

37

It is well known that ride comfort is frequency sensitive. From the ISO2361, the human body is much sensitive to vertical vibrations in the frequency range of 4–8 Hz. Hence, further research work is needed to evaluate the designed constrained suspension in the frequency domain. In the analysis and design of linear systems, the relationship between the time domain and frequency domain characteristics plays an especially important role. However, in system design the state space in the time domain is more convenient as done in the proposed controller. In system analysis, conditions for the robust stability and robust performance are described as specifications on the frequency response in most cases. Therefore, it is indispensable to transform frequency domain conditions into equivalent time domain conditions in order to establish an effective design theory. Parseval's theorem gives the relationship between the squared integral of a time function and that of its Fourier transform, namely, the energy in the time domain is equal to the energy in the frequency domain. As a powerful tool bridging the time domain and frequency domain properties of systems, Kalman-Yakubovich-Popov

passive one as it has a lower body acceleration.

0 0.5 1 1.5 2

time,s

active suspension

road bump-light load

passive suspension

(KYP) lemma plays a fundamental role in robust control.

Case 3: frequency response

Figure 8. Road bump response at light load.

suspension deflection,m

0

0.02

0.04

0.06

0.08

where A and L are the height and the length of the bump and V is the vehicle forward velocity. These values are chosen as A = 0.1 m, L = 5 m, and V = 27 km/h. The bump response, namely, the suspension stroke (m) and the normalized control, is shown for two loads: light and heavy (Figures 8 and 9). In comparison with the passive suspension case, the proposed robust saturated control gives a better dynamic performance in terms of less overshoot and faster damping of the suspension stroke. Moreover, the normalized control signal is bounded between �1 as shown in Figures 8 and 9.

#### 4.1. Passenger comfort

The ride comfort is usually measured by the body acceleration as given in (6). This acceleration is shown in Figure 10.

It is evident from Figures 6–10 that the active suspension outperforms the passive one since it satisfies the control objectives (damps out the oscillations in <1 s and damping ratio > 0.25) as Design of Robust Digital Pole Placer for Car Active Suspension with Input Constraint http://dx.doi.org/10.5772/intechopen.70587 37

Figure 8. Road bump response at light load.

well as it does not violate the constraints of suspension stroke (8 cm) and hydraulic actuator force limit of 1.5 KN. Active suspension also provides better passenger comfort than the passive one as it has a lower body acceleration.

#### Case 3: frequency response

x0 ¼

between �1 as shown in Figures 8 and 9.

4.1. Passenger comfort

Figure 7. Step response at heavy load.

0.005

0.01

0.015

0.02

suspension deflection,m

0.025

0.03

0.035

0.04

0.045

0

is shown in Figure 10.

A

passive suspension

active suspension

8 >><

0 0.5 1

step response

36 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

time,s

>>:

2 1 � cos

2πV L t

normalized control

0

0, t >

, 0 ≤ t ≤

L V

0 0.5 1

step response

time,s

(43)

L V

� � � �

where A and L are the height and the length of the bump and V is the vehicle forward velocity. These values are chosen as A = 0.1 m, L = 5 m, and V = 27 km/h. The bump response, namely, the suspension stroke (m) and the normalized control, is shown for two loads: light and heavy (Figures 8 and 9). In comparison with the passive suspension case, the proposed robust saturated control gives a better dynamic performance in terms of less overshoot and faster damping of the suspension stroke. Moreover, the normalized control signal is bounded

The ride comfort is usually measured by the body acceleration as given in (6). This acceleration

It is evident from Figures 6–10 that the active suspension outperforms the passive one since it satisfies the control objectives (damps out the oscillations in <1 s and damping ratio > 0.25) as It is well known that ride comfort is frequency sensitive. From the ISO2361, the human body is much sensitive to vertical vibrations in the frequency range of 4–8 Hz. Hence, further research work is needed to evaluate the designed constrained suspension in the frequency domain. In the analysis and design of linear systems, the relationship between the time domain and frequency domain characteristics plays an especially important role. However, in system design the state space in the time domain is more convenient as done in the proposed controller. In system analysis, conditions for the robust stability and robust performance are described as specifications on the frequency response in most cases. Therefore, it is indispensable to transform frequency domain conditions into equivalent time domain conditions in order to establish an effective design theory. Parseval's theorem gives the relationship between the squared integral of a time function and that of its Fourier transform, namely, the energy in the time domain is equal to the energy in the frequency domain. As a powerful tool bridging the time domain and frequency domain properties of systems, Kalman-Yakubovich-Popov (KYP) lemma plays a fundamental role in robust control.

5. Conclusion

pension strokes and control inputs within bounds.

Hisham M. Soliman1,2\*, Rashid Al-Abri1 and Mohammed Albadi1

2 Electrical Engineering Department, Cairo University, Cairo, Egypt

dynamics, and control. Vehicle System Dynamics. 2011;49(1–2):3-28

\*Address all correspondence to<EMAIL_ADDRESS>
Vehicle System Dynamics. 2015;53(7):1034-1062

Engineering Practice. 2008;16(12):1457-1467

Identification and Control. 2008;5:66-76

Mechatronics. 2012;17(4):767-778

and frequency domain properties.

Author details

Oman

References

This chapter considers placing the poles in a desired region of uncertain linear digital systems in the presence of input saturation. The new scheme is developed to design robust controllers, taking into consideration the effect of saturation nonlinearity. The proposed controller is based on LMI optimization and requires the uncertainty of model parameters to be cast in the normbounded form. The designed controller is applied to an active suspension of a quarter-vehicle model. Analysis and simulation results have confirmed the potential benefit of the proposed constrained active suspension in achieving the best possible ride comfort while keeping sus-

Design of Robust Digital Pole Placer for Car Active Suspension with Input Constraint

http://dx.doi.org/10.5772/intechopen.70587

39

In the future, it will be interesting to consider the actuator dynamic behavior in the controller design phase and also to develop robust control based on the relation between time domain

1 Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat,

[1] Cao D, Song X, Ahmadian M. Editors's perspectives: Road vehicle suspension design,

[2] Tseng HE, Hrovat D. State of the art survey: Active and semi-active suspension control.

[3] Fallah M, Bhat RB, Xie WF. Optimized control of semiactive suspension systems using H robust control theory and current signal estimation. IEEE/ASME Transactions on

[4] Yagiz N, Hacioglu Y. Backstepping control of a vehicle with active suspensions. Control

[5] Soliman HM, Awadallah MA, Nadim Emira M. Robust controller design for active suspensions using particle swarm optimization. The International Journal of Modeling,

Figure 9. Road bump response at heavy load.

Figure 10. Passenger comfort due to road bump: light load (left) and heavy load (right).

#### 5. Conclusion

This chapter considers placing the poles in a desired region of uncertain linear digital systems in the presence of input saturation. The new scheme is developed to design robust controllers, taking into consideration the effect of saturation nonlinearity. The proposed controller is based on LMI optimization and requires the uncertainty of model parameters to be cast in the normbounded form. The designed controller is applied to an active suspension of a quarter-vehicle model. Analysis and simulation results have confirmed the potential benefit of the proposed constrained active suspension in achieving the best possible ride comfort while keeping suspension strokes and control inputs within bounds.

In the future, it will be interesting to consider the actuator dynamic behavior in the controller design phase and also to develop robust control based on the relation between time domain and frequency domain properties.

### Author details

Hisham M. Soliman1,2\*, Rashid Al-Abri1 and Mohammed Albadi1

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat, Oman

2 Electrical Engineering Department, Cairo University, Cairo, Egypt

#### References

0 0.5 1 1.5 2

0 0.5 1 1.5 2

time,s

passenger comfort,m/s2-heavy load

passive suspension

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.8

time,s

active suspension

normalized control-heavy load

body acceleration,m/s2

normalized control force

0

0.2

0.4

0.6

0.8

1

time,s

active suspension

passenger comfort,m/s2-light load

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1

active suspension

Figure 10. Passenger comfort due to road bump: light load (left) and heavy load (right).

time,s

Figure 9. Road bump response at heavy load.

passive suspension

road bump-heavy load

passive suspension

38 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

body acceleration,m/s2

0.02

suspension deflection,m

0.04

0.06

0.08

0.1

0

[6] Readman MC, Corless M, Villegas C, Shorten R. Adaptive Williams filters for active vehicle suspensions. Transactions of the Institute of Measurement and Control. 2010;32(6):660-676

**Chapter 3**

Provisional chapter

**Bifurcation Makes a Wave Resonant Solid-State Gyro**

DOI: 10.5772/intechopen.71726

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 non-

Keywords: wave solid-state gyro, Foucault pendulum, spring pendulum, bifurcation,

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: cdrFV2=2, where r is the density of a fluid or gas, V is the velocity of a flow, F denotes an effective area of the object, cd

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Bifurcation Makes a Wave Resonant Solid-State

**Be Stable**

Gyro Be Stable

Abstract

stability

1. Introduction

Svetlana Pavlovna Nikitenkova and Dmitry Anatolyevich Kovriguine

Svetlana Pavlovna Nikitenkova and Dmitry Anatolyevich Kovriguine

http://dx.doi.org/10.5772/intechopen.71726

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

linearity of the resonator in a postbifurcation regime.

Provisional chapter

#### **Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable** Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

DOI: 10.5772/intechopen.71726

Svetlana Pavlovna Nikitenkova and Dmitry Anatolyevich Kovriguine Svetlana Pavlovna Nikitenkova and

Additional information is available at the end of the chapter Dmitry Anatolyevich Kovriguine

http://dx.doi.org/10.5772/intechopen.71726 Additional information is available at the end of the chapter

#### Abstract

[6] Readman MC, Corless M, Villegas C, Shorten R. Adaptive Williams filters for active vehicle suspensions. Transactions of the Institute of Measurement and Control. 2010;32(6):660-676

[7] Zuo L, Slotine JJ, Nayfeh SA. Model reaching adaptive control for vibration isolation.

[8] Onat C, Kucukdemiral T, Sivrioglu S, Cansever YG. LPV gain scheduling controller design for a non-linear quarter-vehicle active suspension system. Transactions of the

[9] Chen H, Guo K. Constrained H ∞ control of active suspensions: An LMI approach. IEEE

[10] Suzuki T, Takahashi M. Robust Active Suspension Control for Vibration Reduction of Passenger's Body. Chapter 5; Challenges and Paradigms in Applied Robust Control

[11] Li H, Jing X, Karimi HR. Output-feedback-based H∞ control for vehicle suspension systems with control delay. IEEE Transactions on Industrial Electronics. 2014;61(1):436-446 [12] Li H, Jing X, Lam HK, Shi P. Fuzzy sampled-data control for uncertain vehicle suspension

[13] Li H, Yu J, Hilton C, Liu H. Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T–S fuzzy approach. IEEE Transactions on Industrial Electron-

[14] Akbari A, Lohmann B. Output feedback HN/GH2 preview control of active vehicle suspensions: A comparison study of LQG preview. Vehicle System Dynamics. 2010;48(12):

[15] Mahmoud MS. Decentralized Systems with Design Constraints. London: Springer; 2011 [16] Dorf R, Bishop RH. Modern Control Systems. 11th ed. New Jersey: Prentice Hall; 2008

[17] Haddad WM, Bernstein DS. Controller design with regional pole constraints. IEEE Trans-

[18] Rao PS, Sen I. Robust pole placement stabilizer design using linear matrix inequalities.

[19] Mahmoud MS, Soliman HM. Design of robust power system stabilizer based on particle

[20] Soliman HM, Bajabaa N. Robust guaranteed-cost control with regional pole placement of active suspensions. Journal of Vibration and Control. June 2013;19(8):1170-1186

[22] Soliman HM, Benzaouia A, Yousef H. Saturated robust control with regional pole placement and application to car active suspension. Journal of Vibration and Control. 2016;22(1):

Edited by Andrzej Bartoszewicz. InTech; 2011. ISBN 978-953-307-338-5 472 p

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systems. IEEE Transactions on Cybernetics. 2014;44(7):1111-1126

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ics. August 2013;60(8)

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40 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

Institute of Measurement and Control. 2009;31(1):71-95

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: cdrFV2=2, where r is the density of a fluid or gas, V is the velocity of a flow, F denotes an effective area of the object, cd

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

stands for the so-called drag coefficient being a dimensionless characteristic function versus the Reynolds number. Recall that the Reynolds number is usually defined as Vl=ν, where l is an effective length, ν is the kinematic viscosity coefficient. The function cd is respective for all the uncontrolled points accompanying the interaction between the body and the flow. This function can be recorded experimentally using a wind tube. Experimental results obtained are used to evaluate the drag on other similar objects. In particular, the similarity by the Reynolds number ensures a correct description of viscous motions. Notice that the drag coefficient is nearly a constant in a wide range of Reynolds numbers in most practical cases. As long as the flow velocity would increase, the Reynolds number also increases. It is well known that in a slow flow, the viscosity can be neglected in practice. In this case, one can observe almost an ideal flow with no boundary layer near the surface. Therefore, there is almost no resistance to the motion in accordance with well-known d'Alembert paradox. As the velocity increases further, the drag becomes non-zero as a result of vortex generation. Then, these vortexes oscillate to produce the so-called Kármán vortex street, while the drag increases gradually together with the flow speed. At even higher velocity, the boundary layer turns into a chaotic turbulent flow. It is natural to expect that the turbulent drag would be higher than that in the laminar flow case. Nonetheless, 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. It is observed experimentally that a roughened cylinder or a ball will pass the turbulent flow at a lower Reynolds number than a smooth cylinder or a ball. At first sight, this results in a rather paradoxical result: there is a small range of Reynolds numbers for which the drag of a roughened body is less than the drag of a smooth body of the same size. For a good example, we can recall a golf ball having special dimples. Indeed, the golf with those dimples flies better than an analogous smoothed one [1].

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-pendulumbased 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 mis-

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

http://dx.doi.org/10.5772/intechopen.71726

43

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 [8–10]. 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

where w defines the latitude of geographical place; Ω is the Earth's angular velocity in absolute value; r is the Earth's radius plus a distance from the point where the pendulum is suspended. It is supposed that Earth rotates about the axis z. Here, x tð Þ and y tð Þ denote the pair of projections of infinitesimal oscillations of the bob, weight mg, in the rotating Cartesian frame of referencesð Þ O; x; y; z . The origin O belongs to the Earth's center while the plane ð Þ O; x; z passes through both poles. For operating such a gyro 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

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

y€ þ 2Ω sin wy\_ þ ω2y ¼ �rΩ2 sin w cos w, (1)

x€ � 2Ω sin wx\_ þ ω2x ¼ 0;

energy into these oscillations by external forces for a permanent operating.

sions extending up to 15 years [6].

given by the following two differential equations:

about the sensitivity axis:

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-pendulumbased 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].

stands for the so-called drag coefficient being a dimensionless characteristic function versus the Reynolds number. Recall that the Reynolds number is usually defined as Vl=ν, where l is an effective length, ν is the kinematic viscosity coefficient. The function cd is respective for all the uncontrolled points accompanying the interaction between the body and the flow. This function can be recorded experimentally using a wind tube. Experimental results obtained are used to evaluate the drag on other similar objects. In particular, the similarity by the Reynolds number ensures a correct description of viscous motions. Notice that the drag coefficient is nearly a constant in a wide range of Reynolds numbers in most practical cases. As long as the flow velocity would increase, the Reynolds number also increases. It is well known that in a slow flow, the viscosity can be neglected in practice. In this case, one can observe almost an ideal flow with no boundary layer near the surface. Therefore, there is almost no resistance to the motion in accordance with well-known d'Alembert paradox. As the velocity increases further, the drag becomes non-zero as a result of vortex generation. Then, these vortexes oscillate to produce the so-called Kármán vortex street, while the drag increases gradually together with the flow speed. At even higher velocity, the boundary layer turns into a chaotic turbulent flow. It is natural to expect that the turbulent drag would be higher than that in the laminar flow case. Nonetheless, 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. It is observed experimentally that a roughened cylinder or a ball will pass the turbulent flow at a lower Reynolds number than a smooth cylinder or a ball. At first sight, this results in a rather paradoxical result: there is a small range of Reynolds numbers for which the drag of a roughened body is less than the drag of a smooth body of the same size. For a good example, we can recall a golf ball having special dimples. Indeed, the golf with those dimples flies better

42 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

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

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

than an analogous smoothed one [1].

us briefly touch some key points of this topic.

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 [8–10]. 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:

$$\begin{aligned} \ddot{\mathbf{x}} - 2\boldsymbol{\Omega} \cdot \sin\varphi \dot{\mathbf{x}} + \omega^2 \mathbf{x} &= 0; \\ \ddot{\mathbf{y}} + 2\boldsymbol{\Omega} \cdot \sin\varphi \dot{\mathbf{y}} + \omega^2 \mathbf{y} &= -r\boldsymbol{\Omega}^2 \sin\varphi \cos\varphi, \end{aligned} \tag{1}$$

where w defines the latitude of geographical place; Ω is the Earth's angular velocity in absolute value; r is the Earth's radius plus a distance from the point where the pendulum is suspended. It is supposed that Earth rotates about the axis z. Here, x tð Þ and y tð Þ denote the pair of projections of infinitesimal oscillations of the bob, weight mg, in the rotating Cartesian frame of referencesð Þ O; x; y; z . The origin O belongs to the Earth's center while the plane ð Þ O; x; z passes through both poles. For operating such a gyro 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:

$$\begin{split} \left(\upsilon\_{t,t} - \upsilon\_{\theta,\theta,t,t} - 4\,\omega\upsilon\_{\theta,t} + (\omega^2 - \mu)\upsilon\_{\theta,\theta} + \frac{T(2\upsilon\_{\theta,\theta} + \upsilon\_{\theta,\theta,\theta,\theta})}{\sigma a^2} \right. \\ \left. \tag{2} \end{split} \tag{2}$$

$$= \frac{\beta(\upsilon\_{\theta,\theta,\theta,\theta,\theta,\theta} + 2\upsilon\_{\theta,\theta,t,t} + \upsilon\_{\theta,\theta})}{\sigma a^4}. \tag{3}$$

Alternatively, we may follow the Hamiltonian approach:

p2 1 m2 þ p2

and then writing the following set of Hamiltonian equations:

2 l þ q1 � �2

� � sin q2,

2

arations, it is convenient to pass from the physical coordinates to dimensionless variables:

z4=Ω3

dz2 dτ ¼ z4;

Let us now consider small-but-finite oscillations of the pendulum. In this case, the general

z1ð Þ¼� τ μA1ð Þ T cos ð Þ w1ð Þþ T τ ; z2ð Þ¼� τ μ A2ð Þ T cos ð Þ ν τ þ w2ð Þ T =ν;

z3ð Þ¼ τ μA1ð Þ T sin ð Þ w1ð Þþ T τ ; z4ð Þ¼ τ μA2ð Þ T sin ð Þ ντ þ w2ð Þ T

to look for solutions to Eq. (5) with unknown amplitudes Ai and phases wi slowly evolving in the new slow time T ¼ μτ, where μ is arbitrary small-scaling parameter. The substitution from Eq. (7) into Eq. (5), and ordering in μ, yields the following set of first-order approximation

> 2 2 sin Φ;

32 sin Φ;

16 � 2A2 2 A1 � � cos Φ,

dz4 dτ ¼ �ν2

g=l p denote the natural radial and angular frequencies of the pendulum with an inextensible thread at small oscillations. Notice that the angular frequency would be somewhat

!

; q\_2 ¼ p2 m

l þ Δ þ q1

1 þ Ω2 2 � �2

p\_ 2 ¼ �mg l þ Δ þ q1

2 m Ω2

lower in the case of extensible thread, namely ν ¼ Ω2=

solution to the set (6) can be represented as follows:

8 >>><

>>>:

variables, equations of infinitesimal oscillations would read

dz1 dτ ¼ z3;

dz3 dτ ¼ �z1;

dA1 dT ¼ � A2

dA2

dΦ dT ¼ 9A1

dT ¼ 9A2A1

m2

l þ Δ þ q1 � ��2

� ��3 � kq1 þ mg cos q2 � 1 � �;

þ kq2 1

2 þ mg l þ q1

;

� � 1 � cos q2

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

θ\_ are the canonical coordinates. For further prep-

1; q1 ¼ lz1; q2 ¼ z2, where Ω1 ¼ ffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω2 1 þ Ω2 2

q

z2:

� � � q1 � �, (4)

http://dx.doi.org/10.5772/intechopen.71726

(5)

45

(6)

(7)

(8)

k=m p and

.  The symbol ω denotes the angular rate; a is the radius of the ring; σ is mass density; T ¼ σa2 ω2 � μ � � stands for an attractive force μ, times the distance, directed toward the axis. We can consider this model of the circular ring as a prototype of a solid-state wave gyro to trace the precession of the flexural standing wave 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 T. Nonetheless, some feedback is in need since the parametrically excited oscillations are always unstable either in the presence or in the absence of damping [8]. To maintain the functionality as well as the sensitivity of a conventional wave gyro in practice, the driving of standing waves requires a sophisticated feedback control to eliminate the unstable oscillations caused by their parametric excitation. Therefore, such a kind of wave excitation looks as not quite satisfactory from the viewpoint of actuation, driving or stability properties. Since the linear theory is not able to achieve the desired goal, it is necessary to look for other theoretical tools within the nonlinear wave dynamics. Our thought is to use the exclusive property of the axisymmetric mode of the ring to control the amplitude of flexural waves in the gyro in the presence of energy dissipation, just to use this one as a sort of catalysator in the dynamical process. This chapter demonstrates that when both the primary resonant pumping over the axisymmetric mode and the principal parametric resonant excitation are combined, such a gyro can operate without any feedback, just at the expense of the natural nonlinearity of the resonator in a post-bifurcation regime.

#### 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 m attached to a spring with the stiffness k and of the natural length l. Under the Earth gravity, the equilibrium length of the pendulum becomes l þ Δ, where Δ ¼ mg=k denotes the spring elongation. After introducing the time-dependent radial coordinate r tð Þ, perpendicularly to the pivot of the spring, and the angle θð Þt between the spring and the vertical, the equations governing the motion can be derived easily by using the following Lagrangian function:

$$\mathcal{L} = m \left[ \dot{r}^2 + (l+r)^2 \dot{\theta}^2 - \left( \dot{r}^2 + (l+r)^2 \dot{\theta}^2 \right) / 2 \right] + kr^2/2 + mg[(l+r)1 - \cos\theta - r]. \tag{3}$$

Alternatively, we may follow the Hamiltonian approach:

vt,t � vθ,θ,t,t � 4ωvθ,t þ ω2 � μ � �vθ,θ þ

44 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

natural nonlinearity of the resonator in a post-bifurcation regime.

system. The best candidate seems to be a spring pendulum.

2. Spring pendulum

L ¼ m r\_

2 þ ð Þ l þ r

2 θ\_ 2 � r\_

Here, v ¼ vð Þ θ; t and wð Þ θ; t are the tangential and radial components of displacements measured in the rotating polar frame of references; ð Þ a þ w; θ , at time t. The symbol ω denotes the angular rate; a is the radius of the ring; σ is mass density; T ¼ σa2 ω2 � μ � � stands for an attractive force μ, times the distance, directed toward the axis. We can consider this model of the circular ring as a prototype of a solid-state wave gyro to trace the precession of the flexural standing wave 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 T. Nonetheless, some feedback is in need since the parametrically excited oscillations are always unstable either in the presence or in the absence of damping [8]. To maintain the functionality as well as the sensitivity of a conventional wave gyro in practice, the driving of standing waves requires a sophisticated feedback control to eliminate the unstable oscillations caused by their parametric excitation. Therefore, such a kind of wave excitation looks as not quite satisfactory from the viewpoint of actuation, driving or stability properties. Since the linear theory is not able to achieve the desired goal, it is necessary to look for other theoretical tools within the nonlinear wave dynamics. Our thought is to use the exclusive property of the axisymmetric mode of the ring to control the amplitude of flexural waves in the gyro in the presence of energy dissipation, just to use this one as a sort of catalysator in the dynamical process. This chapter demonstrates that when both the primary resonant pumping over the axisymmetric mode and the principal parametric resonant excitation are combined, such a gyro can operate without any feedback, just at the expense of the

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

Therefore, let us consider a pendulum swinging in a plane and consisting of a bob of mass m attached to a spring with the stiffness k and of the natural length l. Under the Earth gravity, the equilibrium length of the pendulum becomes l þ Δ, where Δ ¼ mg=k denotes the spring elongation. After introducing the time-dependent radial coordinate r tð Þ, perpendicularly to the pivot of the spring, and the angle θð Þt between the spring and the vertical, the equations governing the motion can be derived easily by using the following Lagrangian function:

2 þ ð Þ l þ r

h i

2 θ\_ 2 � �

=2

þ kr2

=2 þ mg l ½ � ð Þ þ r 1 � cos θ � r : (3)

Tð Þ 2vθ,θ þ vθ,θ,θ,θ σa2

σa4 :

(2)

¼ βð Þ vθ,θ,θ,θ,θ,θ þ 2vθ,θ,t,t þ vθ,θ

$$\mathcal{H} = \frac{p\_1^2}{m} + \frac{p\_2^2}{m(l+q\_1)^2} - \frac{1}{2m} \left(\frac{p\_1^2}{m^2} + \frac{p\_2^2}{(l+q\_1)^2 m^2}\right) + \frac{kq\_1^2}{2} + mg\left(\left(l+q\_1\right)\left(1-\cos q\_2\right) - q\_1\right), \tag{4}$$

and then writing the following set of Hamiltonian equations:

$$\begin{aligned} \dot{q}\_1 &= \frac{p\_1}{m}; \quad \dot{q}\_2 = \frac{p\_2}{m} \left(l + \Delta + q\_1\right)^{-2};\\ \dot{p}\_1 &= \frac{p\_2^2}{m} \left(l + \Delta + q\_1\right)^{-3} - kq\_1 + mg\left(\cos q\_2 - 1\right);\\ \dot{p}\_2 &= -mg\left(l + \Delta + q\_1\right)\sin q\_{2'} \end{aligned} \tag{5}$$

where q1 ¼ r, q2 ¼ θ, p1 ¼ mr\_, p2 ¼ m lð Þ þ r 2 θ\_ are the canonical coordinates. For further preparations, it is convenient to pass from the physical coordinates to dimensionless variables: t ¼ τ=Ω1; p1 ¼ lmΩ1 z3; p2 ¼ l 2 m Ω2 1 þ Ω2 2 � �2 z4=Ω3 1; q1 ¼ lz1; q2 ¼ z2, where Ω1 ¼ ffiffiffiffiffiffiffiffi k=m p and Ω2 ¼ ffiffiffiffiffiffi g=l p denote the natural radial and angular frequencies of the pendulum with an inextensible thread at small oscillations. Notice that the angular frequency would be somewhat lower in the case of extensible thread, namely ν ¼ Ω2= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω2 1 þ Ω2 2 q . In the new dimensionless variables, equations of infinitesimal oscillations would read

$$\begin{aligned} \frac{dz\_1}{d\tau} &= z\_3; \quad \frac{dz\_2}{d\tau} = z\_4; \\ \frac{dz\_3}{d\tau} &= -z\_1; \quad \frac{dz\_4}{d\tau} = -\nu^2 z\_2. \end{aligned} \tag{6}$$

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:

$$\begin{cases} z\_1(\tau) = -\mu A\_1(T) \cos \ (\varphi\_1(T) + \tau); \\ z\_2(\tau) = -\mu \ A\_2(T) \cos \ (\nu \tau + \varphi\_2(T)) / \nu; \\ z\_3(\tau) = \mu A\_1(T) \sin \ (\varphi\_1(T) + \tau); \\ z\_4(\tau) = \mu A\_2(T) \sin \ (\nu \tau + \varphi\_2(T)) \end{cases} \tag{7}$$

to look for solutions to Eq. (5) with unknown amplitudes Ai and phases wi slowly evolving in the new slow time T ¼ μτ, where μ is arbitrary small-scaling parameter. The substitution from Eq. (7) into Eq. (5), and ordering in μ, yields the following set of first-order approximation evolution equations:

$$\begin{aligned} \frac{dA\_1}{dT} &= -\frac{A\_2^2}{2} \sin \Phi;\\ \frac{dA\_2}{dT} &= \frac{9A\_2A\_1}{32} \sin \Phi;\\ \frac{d\Phi}{dT} &= \left(\frac{9A\_1}{16} - \frac{2A\_2^2}{A\_1}\right) \cos \Phi,\end{aligned} \tag{8}$$

under the one assumption that the following integer-valued ratio between the frequencies, manifesting the primary parametric resonance, that is, 1 : ν ¼ 1 : 2, takes place. Notice that the set (8) contains the so-called generalized phase Φð Þ¼ T w1ð Þ� T 2w2ð Þ T , as the unknown variable. The set (8) is of Hamiltonian structure. So, the related average Hamiltonian function reads

$$A\_2^2(T)A\_1(T)\cos\Phi(T) = A\_2^2(0)A\_1(0). \tag{9}$$

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

$$\mathcal{E} = \frac{9A\_1^2}{32} + \frac{A\_2^2}{2},\tag{10}$$

�A2

having the following solutions:

and

2=2 þ G � ΛA1 ¼ 0;

A1 ¼ 32Λ

way to the new stationary steady state (15).

A1 ¼ G

9 , A2 ¼

stationary solution (14) near the bifurcation point K ¼ 32Λ2

9A2A1

Λ , A2 ¼ 0, when K ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18G � 64Λ2 p

These solutions, Eqs. (14) and (15), are plotted in Figure 1. As we can see, the dimensionless parameter K plays the role of control parameter governing the bifurcation of the system. The

Figure 1. Bifurcation diagram—Amplitudes of radial and angular modes versus control parameter K. Solid circles refer to stable angular oscillations, circles mark stable radial oscillations. Crosses denote unstable radial mode. Change in the

2

p has been scaled arbitrary for the best view).

stability takes place near the bifurcation point K (parameter Λ ¼ 3=8 ffiffiffi

32 � ΛA2 ¼ 0; Φð Þ¼ T π=2, (13)

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

32Λ2

9 ; (14)

http://dx.doi.org/10.5772/intechopen.71726

47

9 : (15)

=9 becomes unstable and gives

32Λ2

=3, as K >

where E ¼ 9A2 1ð Þ0 =32 þ A2 2ð Þ0 =2 is the average kinetic energy of the pendulum defined at the initial instant of time. These two integrals, Eqs. (9) and (10), allow us to integrate the set (8) analytically in terms of Jacobi elliptic functions [3].

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:

$$\begin{aligned} \dot{q}\_1 &= \frac{p\_1}{m}; \quad \dot{q}\_2 = \frac{p\_2}{m} \left(l + \Delta + q\_1\right)^{-2};\\ \dot{p}\_1 &= \frac{p\_2^2}{m} \left(l + \Delta + q\_1\right)^{-3} - kq\_1 + m\mathfrak{g}\left(\cos q\_2 - 1\right) - 2\mu\delta p\_1 - \mu F \cos\left(\Omega\_1 t + \psi\_1\right);\\ \dot{p}\_2 &= -m\mathfrak{g}\left(l + \Delta + q\_1\right)\sin q\_2 - 2\mu\delta p\_2. \end{aligned} \tag{11}$$

where δ is the viscous drag coefficient; F is the amplitude of an external harmonic force acting at the resonant frequency of free radial oscillations of the pendulum. We can utilize the same anzats (7) to derive the truncated set of modulation equation for slowly varying amplitudes and phases. Small parameter μ emphasizes that both the damping and forcing are small but finite. After the substitution from Eq. (7) into Eq. (11), and ordering in μ, the first-order approximation evolution equations describing the resonant excitation of the pendulum over the radial mode can be written as

$$\begin{aligned} \frac{dA\_1}{dT} &= -A\_2^2 \sin \Phi / 2 + G - \Lambda A\_1; \\ \frac{dA\_2}{dT} &= \frac{9A\_2 A\_1 \sin \Phi}{32} - \Lambda A\_2; \\ \frac{d\Phi}{dT} &= -\left(\frac{A\_2^2}{2A\_1} + \frac{9A\_1}{16}\right) \cos \Phi. \end{aligned} \tag{12}$$

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, �w1ð Þþ T ψ1 ¼ π=2, accordingly the phase matching conditions. Notice that the parameter G ¼ F=6mg, entering Eq. (12), is interpreted as the dimensionless force, while Λ ¼ δ=Ω1 stands for the dimensionless drag coefficient.

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

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable http://dx.doi.org/10.5772/intechopen.71726 47

$$-A\_2^2/2 + G - \Lambda A\_1 = 0; \quad \frac{9A\_2A\_1}{32} - \Lambda A\_2 = 0; \quad \Phi(T) = \pi/2,\tag{13}$$

having the following solutions:

$$A\_1 = \frac{G}{\Lambda}, A\_2 = 0, \quad \text{when } K \le \frac{32\Lambda^2}{9};\tag{14}$$

and

under the one assumption that the following integer-valued ratio between the frequencies, manifesting the primary parametric resonance, that is, 1 : ν ¼ 1 : 2, takes place. Notice that the set (8) contains the so-called generalized phase Φð Þ¼ T w1ð Þ� T 2w2ð Þ T , as the unknown variable. The set (8) is of Hamiltonian structure. So, the related average Hamiltonian function reads

2ð Þ0 A1ð Þ0 : (9)

2 , (10)

;

(11)

(12)

2ð Þ T A1ð Þ T cos Φð Þ¼ T A2

E ¼ 9A1

;

2 32 þ

initial instant of time. These two integrals, Eqs. (9) and (10), allow us to integrate the set (8)

Let us now refer to the damped forced motion of the pendulum. In this case, equations

�3 � kq1 þ mg cos q2 � 1 � 2μδp1 � μF cos Ω1t þ ψ1

2 sin Φ=2 þ G � ΛA1;

cos Φ:

32 � ΛA2;

where δ is the viscous drag coefficient; F is the amplitude of an external harmonic force acting at the resonant frequency of free radial oscillations of the pendulum. We can utilize the same anzats (7) to derive the truncated set of modulation equation for slowly varying amplitudes and phases. Small parameter μ emphasizes that both the damping and forcing are small but finite. After the substitution from Eq. (7) into Eq. (11), and ordering in μ, the first-order approximation evolution equations describing the resonant excitation of the pendulum over

A2 2

2ð Þ0 =2 is the average kinetic energy of the pendulum defined at the

A2

where E ¼ 9A2

q\_ 1 ¼ p1 m

p\_ 1 ¼ p2 2 m

1ð Þ0 =32 þ A2

; q\_2 ¼ p2 m

l þ Δ þ q1

p\_ 2 ¼ �mg l þ Δ þ q1

the radial mode can be written as

following algebraic set of equations:

analytically in terms of Jacobi elliptic functions [3].

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

46 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

governing the motion (5) can be easily modified to the following form:

l þ Δ þ q1 �2

sin q2 � 2μδp2,

dA1 dT ¼ �A2

dA2

dΦ

dT ¼ 9A2A1 sin Φ

dimensionless force, while Λ ¼ δ=Ω1 stands for the dimensionless drag coefficient.

2 2A1 þ 9A1 16

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, �w1ð Þþ T ψ1 ¼ π=2, accordingly the phase matching conditions. Notice that the parameter G ¼ F=6mg, entering Eq. (12), is interpreted as the

In the case of stationary motion, when the amplitudes are constants, Eq. (12) produces the

dT ¼ � A2

$$A\_1 = \frac{32\Lambda}{9}, A\_2 = \sqrt{18G - 64\Lambda^2}/3, \quad \text{as } K > \frac{32\Lambda^2}{9}. \tag{15}$$

These solutions, Eqs. (14) and (15), are plotted in Figure 1. As we can see, the dimensionless parameter K plays the role of control parameter governing the bifurcation of the system. The stationary solution (14) near the bifurcation point K ¼ 32Λ2 =9 becomes unstable and gives way to the new stationary steady state (15).

Figure 1. Bifurcation diagram—Amplitudes of radial and angular modes versus control parameter K. Solid circles refer to stable angular oscillations, circles mark stable radial oscillations. Crosses denote unstable radial mode. Change in the stability takes place near the bifurcation point K (parameter Λ ¼ 3=8 ffiffiffi 2 p has been scaled arbitrary for the best view).

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):

$$\frac{d}{dT}\left(\frac{9A\_1^2}{16} + A\_2^2\right) = -\Lambda \left(\frac{9A\_1^2}{16} + A\_2^2\right) + \frac{9GA\_1}{16}.\tag{16}$$

follows: F ¼ 1:0 N; β ¼ 0:01 kg=s; δ ¼ 0:01 s�1; g ¼ 9:8 m=s2; k ¼ 29:40 kg=s2; l ¼ 1:0 m;

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 zið Þτ cannot play any role at large times, though we have decided to start in our calculations from the static equilibrium point. Figure 3(a) displays the time history of the radial coordinate of the system governed by Eq. (11), rewritten in new dimensionless variables zið Þτ , while the evolution of the angular oscillations is shown in Figure 3(b). Figure 4 displays a typical Lissajous curve that appears on ð Þ z1; z2 cross section of the four-dimensional phase space for the fully developed stationary state. Figure 5 shows individual cross sections related to the radial and angular vibrations in the developed stationary regime of oscillations. Finally, Figure 6 exhibits the evolution of the Hamiltonian function (4) or the energy of the system (11) in comparison with that one where the angular coordinate is fixed to zero but the force excitation remains the same. Notice that if the angular coordinate is fixed, then the pendulum just performs linear

In new dimensionless variables zið Þτ , the energy of the oscillating pendulum is given by the

cos z2 þ 81z4

1 þ 162z2

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

Figure 3. The evolution of the damped forced radial coordinate of the pendulum in time τ (a) and the time history of the

1 þ 162z3

4 þ 54 !: (17)

3 þ 108 � �z1 þ 81 z2

1þ

(a) (b)

3 þ 256z2

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

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49

�54ð Þ z1 þ 1 3

resonator in more general case (for instance, see [4, 5] and references therein).

81z2 3 þ 135 � �z2

resonantly excited angular coordinate of the pendulum in time τ (b).

m ¼ 1:0 kg.

oscillations in the radial direction.

1 18ð Þ z1 þ 1 2

following explicit expression:

Eð Þ¼ τ

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 K is shown in Figure 2. As we can see, the power of drag forces reduces after the bifurcation under the same external harmonic excitation. This means stability.

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: t ¼ τ ffiffiffiffiffiffiffiffi m=k p ; p1ðÞ¼ t lm ffiffiffiffiffiffiffiffi k=m p z3ð Þτ ; p2ðÞ¼ t m ffiffiffiffiffiffiffiffi k=m p ð Þ mg þ lk 2 z4ð Þτ =k 2 ; q1ðÞ¼ t lz1ð Þτ ; q2ðÞ¼ t z2ð Þτ , from the physical to dimensionless variables. The numerical parameters of the system are chosen by us as

Figure 2. Bifurcation diagram—Average kinetic energy versus control parameter K. Solid lines refer to stable oscillations while dotted line marks unstable regime of oscillations. Parameter Λ is the same as in Figure 1.

follows: F ¼ 1:0 N; β ¼ 0:01 kg=s; δ ¼ 0:01 s�1; g ¼ 9:8 m=s2; k ¼ 29:40 kg=s2; l ¼ 1:0 m; m ¼ 1:0 kg.

To examine the stability properties without any excess mathematical preparations, let us

¼ �Λ 9A2

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 K is shown in Figure 2. As we can see, the power of drag forces reduces after the bifurcation under the same external

Let us now refer to the original Eq. (11) to verify results yielded by the modulation theory. To

to dimensionless variables. The numerical parameters of the system are chosen by us as

Figure 2. Bifurcation diagram—Average kinetic energy versus control parameter K. Solid lines refer to stable oscillations

while dotted line marks unstable regime of oscillations. Parameter Λ is the same as in Figure 1.

m=k p ; p1ðÞ¼ t lm ffiffiffiffiffiffiffiffi

plot the numerical results, we have used the following transform: t ¼ τ ffiffiffiffiffiffiffiffi

z4ð Þτ =k 2

1 16 þ A2 2

� �

þ 9GA1

; q1ðÞ¼ t lz1ð Þτ ; q2ðÞ¼ t z2ð Þτ , from the physical

16 : (16)

analyze the power of drag forces using the energy balance resulted from the set (12):

d dT

harmonic excitation. This means stability.

k=m p z3ð Þτ ; p2ðÞ¼ t m ffiffiffiffiffiffiffiffi

9A2 1 16 þ A2 2

k=m p ð Þ mg þ lk 2

� �

48 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

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 zið Þτ cannot play any role at large times, though we have decided to start in our calculations from the static equilibrium point. Figure 3(a) displays the time history of the radial coordinate of the system governed by Eq. (11), rewritten in new dimensionless variables zið Þτ , while the evolution of the angular oscillations is shown in Figure 3(b). Figure 4 displays a typical Lissajous curve that appears on ð Þ z1; z2 cross section of the four-dimensional phase space for the fully developed stationary state. Figure 5 shows individual cross sections related to the radial and angular vibrations in the developed stationary regime of oscillations. Finally, Figure 6 exhibits the evolution of the Hamiltonian function (4) or the energy of the system (11) in comparison with that one where the angular coordinate is fixed to zero but the force excitation remains the same. Notice that if the angular coordinate is fixed, then the pendulum just performs linear oscillations in the radial direction.

In new dimensionless variables zið Þτ , the energy of the oscillating pendulum is given by the following explicit expression:

$$E(\tau) = \frac{1}{18(z\_1+1)^2} \begin{pmatrix} -54(z\_1+1)^3 \cos z\_2 + 81z\_1^4 + 162z\_1^3 + \\ \left(81z\_3^2 + 135\right)z\_1^2 + \left(162z\_3^2 + 108\right)z\_1 + 81z\_3^2 + 256z\_4^2 + 54 \end{pmatrix} . \tag{17}$$

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).

Figure 3. The evolution of the damped forced radial coordinate of the pendulum in time τ (a) and the time history of the resonantly excited angular coordinate of the pendulum in time τ (b).

Figure 4. The evolution of the damped forced radial coordinate of the pendulum in time τ.

coordinates in the absolute Newton's space ð Þ R þ w;f þ v=R and in the rotating plane ð Þ R þ w; w þ v=R where both have the same center as pole. Here, w and v are radial and

Figure 6. Energy of the pendulum becomes some lower in comparison with the same system with fixed angular

f ¼ w þ Θt,

if the ring rotates uniformly in time t. Also, we may suppose that the angular rate as a slowly varying function of time. In this case, the latter expression is slightly modified by the following

ðt

Θð Þτ dτ:

2R2 v2 � 1

2 ð h=2

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

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�h=2

Ee2 ssdζ,

t0

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 uð Þs ¼ v � ζð Þ ws � v=R and uð Þζ ¼ w, where v ¼ v sð Þ ; t and w ¼ w sð Þ ; t are the components of displacements, rewritten as functions of the circumferential coordinate s; ζ is the distance from the mid-line along the radius.

f ¼ w þ

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

2 h i � EFκ2

2 ð Þ vt þ Θw þ RΘ 2 þ ð Þ wt � Θv

tangential displacements of the ring, respectively. Therefore, we shall have

coordinate. Dashed line refers to a spring pendulum with constrained to zero angular coordinate.

integral:

L ¼ rF

Figure 5. Cross sections of the phase space demonstrating individual dynamical behavior over the radial (a) and angular coordinates (b) at the damped forced oscillations in a spring pendulum in time τ.

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 h and radius R, rotating with angular velocity Θ about its sensitive axis, we introduce the two frames of polar

Figure 6. Energy of the pendulum becomes some lower in comparison with the same system with fixed angular coordinate. Dashed line refers to a spring pendulum with constrained to zero angular coordinate.

coordinates in the absolute Newton's space ð Þ R þ w;f þ v=R and in the rotating plane ð Þ R þ w; w þ v=R where both have the same center as pole. Here, w and v are radial and tangential displacements of the ring, respectively. Therefore, we shall have

$$
\phi = \varphi + \Theta t\_\prime
$$

if the ring rotates uniformly in time t. Also, we may suppose that the angular rate as a slowly varying function of time. In this case, the latter expression is slightly modified by the following integral:

$$\phi = \varphi + \int\_{t\_0}^{t} \Theta(\tau)d\tau.$$

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 uð Þs ¼ v � ζð Þ ws � v=R and uð Þζ ¼ w, where v ¼ v sð Þ ; t and w ¼ w sð Þ ; t are the components of displacements, rewritten as functions of the circumferential coordinate s; ζ is the distance from the mid-line along the radius.

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

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 consider-

Figure 5. Cross sections of the phase space demonstrating individual dynamical behavior over the radial (a) and angular

(a) (b)

In order to define the in-plane position in a thin-ring resonator, thickness h and radius R, rotating with angular velocity Θ about its sensitive axis, we introduce the two frames of polar

ation, namely a thin-ring resonator of which is described in detail in [11].

coordinates (b) at the damped forced oscillations in a spring pendulum in time τ.

Figure 4. The evolution of the damped forced radial coordinate of the pendulum in time τ.

50 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

3. Thin-ring resonator

$$\mathcal{L} = \frac{\rho F}{2} \left[ \left( v\_t + \Theta w + R\Theta \right)^2 + \left( w\_t - \Theta v \right)^2 \right] - \frac{E F \kappa^2}{2 \mathcal{R}^2} v^2 - \frac{1}{2} \int\_{-h/2}^{h/2} E \mathbf{e}\_{ss}^2 d\zeta\_{ss}$$

where E is Young's modulus; F denotes cross-section square; r is mass density; κ characterizes the stiffness of the rotating platform of the ring; ess ¼ vs þ w=R þ ξð Þþ ws,s � vs=R w2 s =2 is the circumferential component of the deformation tensor. According to paradigms of the variational analysis, equations governing the motion in the ring have the following form:

$$\begin{aligned} \left(\mathcal{L}\_{\vec{v}}\right)\_t + \left(\mathcal{L}\_{v\_\*}\right)\_s - \mathcal{L}\_v &= Q\_{(v)} - \mathcal{R}\_{\vec{v}};\\ \left(\mathcal{L}\_{\vec{w}}\right)\_t + \left(\mathcal{L}\_{w\_\*}\right)\_s - \left(\mathcal{L}\_{w\_{\*,\*}}\right)\_{s,s} - \mathcal{L}\_w &= Q\_{(w)} - \mathcal{R}\_{\vec{w}}.\end{aligned} \tag{18}$$

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

Here, we can trace the appearance of the constant extension in radial direction caused by the centrifuge force. Notice that the amplitudes, A and B, entering therein, are linearly interrelated, that is, B ¼ pA, through the interrelation coefficients, p, defined for both the high- and low-

> pk,n ¼ � i 2μΩωk,n þ n 1 þ e2n2 n2 1 þ e2 ð Þþ κ2 � ω2

The high- and low-frequency branches are indexed by k ¼ 1 and k ¼ 2, correspondingly. Recall from the linear algebra that these coefficients should satisfy the orthogonality condition, that is, p1,np2,n ¼ �1, for arbitrary wave number n, excluding the one case of linearly decoupled oscillations, taking place at n ¼ 0, that is, at axisymmetric radial oscillations. The natural

n2 1 þ e2 þ κ2 � ω2 þ μ2Ω2 1 þ e2n4 � ω2 þ μ2Ω2 � 2μΩω � n 1 þ e2n2 2 ¼ 0:

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

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

In the case of a small angular rate, that is, Ω ≪ ωk,n, the roots of the dispersion relation (22) are

It may not, perhaps, be out of place to note that these roots possess by asymmetry because of the angular rate. Nonetheless, each frequency, ωk,n, can be decomposed on the antisymmetric term as well as on the symmetric part. The antisymmetric term is respective for the wave precession caused by the ring rotation while the symmetric one describes a small inessential correction.

2μΩn 1 þ e2n2

1 þ e2n4 þ 1 þ e2 ð Þn2 þ κ2 � 2ð Þ ωk,nð Þ0 2 : (24)

�n6e2 � e2 �ω2 � 2 þ κ2 n4 þ ω2 � 1 e2 þ ω2 n2 � ω4 þ 1 þ κ2 ω2 � κ2 ¼ 0, (23)

frequencies of waves, ωk,n, are defined by the dispersion relation

e μ2Ω2 � 1 þ B exp i nð Þ w þ ω τ ; wð Þ¼ τ; w A exp i nð Þ w þ ω τ ,

k,n þ μ2Ω2 : (21)

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(22)

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

arbitrary, but moderate, angular acceleration [8].

vð Þ¼� τ; w μΩ2

frequency sets of normal modes:

the circumferential modes of extension.

approximately represented as follows:

ωk,nð Þ Ω ≈ ωk,nð Þþ 0

equation

where Qð Þv and Qð Þ w are introduced as generalized forces; the function R ¼ ηK, expressed through the kinetic energy K, is respective for the energy dissipation model linearly scaled by the coefficient η. For further preparations, it seems to be continent to rearrange these equations to dimensionless notation:

$$\begin{aligned} &\ddot{v} + 2\mu\Omega \dot{w} + \mu \dot{\Omega} w + \dot{\Omega} / \epsilon - \mu^2 \Omega^2 v - V\_{\varphi} + \epsilon^2 W\_{\varphi, \varphi} + \kappa^2 v = \frac{\epsilon \mu}{2} \left( w\_{\varphi}^2 \right)\_{\varphi} + Q\_{\psi} \dot{\nu} \\ &\ddot{w} - 2\mu\Omega \dot{v} - \mu \dot{\Omega} v - \mu^2 \Omega^2 w - \mu \Omega^2 / \epsilon + V + \epsilon^2 W\_{\varphi, \varphi, \varphi} \\ &= \epsilon \mu \left[ \left( V w\_{\varphi} \right)\_{\varphi} + \frac{w\_{\varphi}^2}{2} \right] + \frac{\epsilon^2 \mu^2}{2} \left( w\_{\varphi}^3 \right)\_{\varphi} + Q\_{w\nu} \end{aligned} \tag{19}$$

where e ¼ h= ffiffiffiffiffi 12 p R ≪ 1 is the relative thickness of the ring; small parameter μ ¼ a=R ≪ 1 is intended for making up further procedures of the perturbation analysis; the functions V ¼ vw þ w and W ¼ ww � v are written for brevity. The transform to these dimensionless variables reads: vð Þ! τ; w v sð Þ ; t =μa; wð Þ! τ; w w sð Þ ; t =μa; w ¼ s=R; τ ¼ tc=R; Ωð Þ¼ τ Θð Þt R= μc. Here, c ¼ ffiffiffiffiffiffiffiffi E=r p denotes the wave propagation velocity; a ¼ h= ffiffiffiffiffi 12 p .

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

$$\upsilon(\varphi,\tau) = \upsilon(\varphi + 2\pi,\tau); \quad w(\varphi,\tau) = w(\varphi + 2\pi,\tau). \tag{20}$$

#### 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

where E is Young's modulus; F denotes cross-section square; r is mass density; κ characterizes

circumferential component of the deformation tensor. According to paradigms of the varia-

where Qð Þv and Qð Þ w are introduced as generalized forces; the function R ¼ ηK, expressed through the kinetic energy K, is respective for the energy dissipation model linearly scaled by the coefficient η. For further preparations, it seems to be continent to rearrange these equations

v � Vw þ e2Ww,w þ κ2v ¼ eμ

=e þ V þ e2Ww,w,w

p R ≪ 1 is the relative thickness of the ring; small parameter μ ¼ a=R ≪ 1 is

vð Þ¼ w; τ vð Þ w þ 2π; τ ; wð Þ¼ w; τ wð Þ w þ 2π; τ : (20)

w þ Qw,

intended for making up further procedures of the perturbation analysis; the functions V ¼ vw þ w and W ¼ ww � v are written for brevity. The transform to these dimensionless variables reads: vð Þ! τ; w v sð Þ ; t =μa; wð Þ! τ; w w sð Þ ; t =μa; w ¼ s=R; τ ¼ tc=R; Ωð Þ¼ τ Θð Þt R=

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

E=r p denotes the wave propagation velocity; a ¼ h= ffiffiffiffiffi

s =2 is the

(19)

s,s � Lw ¼ Qð Þ w � Rw\_ , (18)

2 w2 w � �

12 p . w þ Qv;

the stiffness of the rotating platform of the ring; ess ¼ vs þ w=R þ ξð Þþ ws,s � vs=R w2

tional analysis, equations governing the motion in the ring have the following form:

� �

ð Þ Lv\_ t þ Lvs ð Þs � Lv ¼ Qð Þv � Rv\_ ;

w � μΩ2

þ e2μ2 2 w3 w � �

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

ð Þ Lw\_ t þ Lws ð Þs � Lws,s

52 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

v€ þ 2μΩw\_ þ μΩ\_ w þ Ω\_ =e � μ2Ω2

w þ w2 w 2

" #

w€ � 2μΩv\_ � μΩ\_ v � μ2Ω2

¼ eμ Vww � �

12

to dimensionless notation:

where e ¼ h= ffiffiffiffiffi

μc. Here, c ¼ ffiffiffiffiffiffiffiffi

3.1. Dispersion relation

$$w(\tau,\varphi) = -\frac{\mu\,\Omega^2}{\epsilon\left(\mu^2\Omega^2 - 1\right)} + B\exp\left\|(n\varphi + \omega\tau)\right\|; \quad w(\tau,\varphi) = A\exp\left\|(n\varphi + \omega\tau)\right\|$$

Here, we can trace the appearance of the constant extension in radial direction caused by the centrifuge force. Notice that the amplitudes, A and B, entering therein, are linearly interrelated, that is, B ¼ pA, through the interrelation coefficients, p, defined for both the high- and lowfrequency sets of normal modes:

$$p\_{k,n} = -\frac{i\left(2\,\mu\,\Omega\,\omega\_{k,n} + n\left(1 + \epsilon^2 n^2\right)\right)}{n^2(1+\epsilon^2) + \kappa^2 - \left(\omega\_{k,n}^2 + \mu^2 \Omega^2\right)}.\tag{21}$$

The high- and low-frequency branches are indexed by k ¼ 1 and k ¼ 2, correspondingly. Recall from the linear algebra that these coefficients should satisfy the orthogonality condition, that is, p1,np2,n ¼ �1, for arbitrary wave number n, excluding the one case of linearly decoupled oscillations, taking place at n ¼ 0, that is, at axisymmetric radial oscillations. The natural frequencies of waves, ωk,n, are defined by the dispersion relation

$$\left(\mu^2 \left(1 + \epsilon^2\right) + \kappa^2 - \left(\omega^2 + \mu^2 \Omega^2\right)\right) \left(1 + \epsilon^2 n^4 - \left(\omega^2 + \mu^2 \Omega^2\right)\right) - \left(2\mu \Omega \omega - n \left(1 + \epsilon^2 n^2\right)\right)^2 = 0. \tag{22}$$

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

$$-n^6\epsilon^2 - \epsilon^2(-\omega^2 - 2 + \kappa^2)n^4 + \left((\omega^2 - 1)\epsilon^2 + \omega^2\right)n^2 - \omega^4 + (1 + \kappa^2)\omega^2 - \kappa^2 = 0,\tag{23}$$

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, Ω ≪ ωk,n, the roots of the dispersion relation (22) are approximately represented as follows:

$$
\omega\_{k,n}(\Omega) \approx \omega\_{k,n}(0) + \frac{2\mu\Omega n \left(1 + \epsilon^2 n^2\right)}{1 + \epsilon^2 n^4 + (1 + \epsilon^2)n^2 + \kappa^2 - 2\left(\omega\_{k,n}(0)\right)^2}.\tag{24}
$$

It may not, perhaps, be out of place to note that these roots possess by asymmetry because of the angular rate. Nonetheless, each frequency, ωk,n, can be decomposed on the antisymmetric term as well as on the symmetric part. The antisymmetric term is respective for the wave precession caused by the ring rotation while the symmetric one describes a small inessential correction.

We assume that the angular rate and acceleration of the ring are small enough, that is, Θð Þ� τ μ and dΘ=dτ � μ2. Also, we should suppose that spectral parameters of waves in the

where Δω denotes a small discrepancy between the phases of the high-frequency axisymmetric radial oscillation and two low-frequency quasi-harmonic waves with amplitudes, numbers one and two. Certainly, all the frequencies ω1, ω2, ω3 and the wave numbers �n should satisfy the dispersion relation (23). In the theory of nonlinear waves, such a trio is used to name a 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

ð2π 0

Ldf1 � �df2

L2 þ …,

∂L1 ∂Ak,T

� L1,

2 þ en2 A1A2A3exp ð Þþ iΔωT A1A2A3exp ð Þ �iΔωT � �: (28)

ω 1 þ p2 ð Þ A2A3eiΔωT

ω 1 þ p2 ð Þ A1A3eiΔωT;

(29)

ð2π 0

ω3 ¼ ω1 þ ω2 þ μΔω, (26)

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

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55

� �df3, (27)

solution (25) are matched by the following phase synchronicity conditions:

Hamiltonian formalism. Let us consider the average Lagrangian of the system

0

in order than to expand this one in a formal series in small parameter μ as follows:

h i L ¼ L0 þ μL1 þ μ2

∂L1 ∂Ak,T

Here, zero-order approximation term, that is, L0 ¼ 0, would coincide exactly with the dispersion relation, accordingly to findings of Whitham [16]. The second term L1, entering this expansion, describes effects arising in the first-order nonlinear approximation analysis which we need to study here. Then, the nontrivial average Hamiltonian of the system would read

þ Ak,T

!

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

Recall that the bending wave frequency ω ¼ ω1 ¼ ω2 should be close to the frequency ω3=2 that denotes the frequency of the axisymmetric radial oscillation, that is, ω3 ¼ 1. Note that the case under consideration is associated with the principal parametric resonance. His Hamiltonian

1 þ p2 A1 � ien2

1 þ p2 A1 � ie n2

A1A2e�iΔωT,

h i L ¼ 1

H ¼ X 3

pragmatic form

H ¼ 2iΩX

2

j¼1

ωj pj � pj � � Aj � � � �

produces a set of following evolution equations:

dA1

dA2

dA3

dT ¼ 2Ω p1 � p1

dT ¼ 2Ω p2 � p2

dT ¼ �ie n2

� �

� �

k¼1

Ak,T

2π � �3 ð2π

Figure 7. The high- and low-frequency dispersion branches for waves on the fixed platform (e ¼ 0:01, κ ¼ 0:45, Ω ¼ 0).

#### 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 [12–15]:

$$w(\tau,\varphi) = \sum\_{k=1}^{3} p\_k A\_k(T) \exp i\phi\_k; \quad w(\tau,\varphi) = \sum\_{k=1}^{3} A\_k(T) \exp i\phi\_{k\prime} \tag{25}$$

which represent an approximated solution to Eq. (19). Here, Akð Þ T (k ¼ 1::3) are the slowly varying complex wave amplitudes, evolving at slow time T ¼ μτ; f1 ¼ nw þ ω1τ, f2 ¼ �nw þ ω1τ, f3 ¼ ω3τ are the fast rotating phases; ∗ð Þ denote the complex conjugation. Notice that the amplitude and phase number three refer to the axisymmetric radial oscillation of the ring. We assume that the angular rate and acceleration of the ring are small enough, that is, Θð Þ� τ μ and dΘ=dτ � μ2. Also, we should suppose that spectral parameters of waves in the solution (25) are matched by the following phase synchronicity conditions:

$$
\omega\_3 = \omega\_1 + \omega\_2 + \mu \Delta \omega\_\prime \tag{26}
$$

where Δω denotes a small discrepancy between the phases of the high-frequency axisymmetric radial oscillation and two low-frequency quasi-harmonic waves with amplitudes, numbers one and two. Certainly, all the frequencies ω1, ω2, ω3 and the wave numbers �n should satisfy the dispersion relation (23). In the theory of nonlinear waves, such a trio is used to name a 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

$$
\langle \mathcal{L} \rangle = \left(\frac{1}{2\pi}\right)^3 \int\_0^{2\pi} \left( \int\_0^{2\pi} \mathcal{L} d\phi\_1 \right) d\phi\_2 \right) d\phi\_3 \tag{27}
$$

in order than to expand this one in a formal series in small parameter μ as follows:

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

54 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

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

vð Þ¼ τ; w X

3

k¼1

tudes and phases can be directly obtained using the following anzats [12–15]:

solid-state wave gyro.

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

Figure 7. The high- and low-frequency dispersion branches for waves on the fixed platform (e ¼ 0:01, κ ¼ 0:45, Ω ¼ 0).

Truncated equations or evolution equations describing the modulation phenomena of ampli-

pkAkð Þ T exp ifk; wð Þ¼ τ; w X

which represent an approximated solution to Eq. (19). Here, Akð Þ T (k ¼ 1::3) are the slowly varying complex wave amplitudes, evolving at slow time T ¼ μτ; f1 ¼ nw þ ω1τ, f2 ¼ �nw þ ω1τ, f3 ¼ ω3τ are the fast rotating phases; ∗ð Þ denote the complex conjugation. Notice that the amplitude and phase number three refer to the axisymmetric radial oscillation of the ring.

3

k¼1

Akð Þ T exp ifk, (25)

$$
\langle \mathcal{L} \rangle = \mathcal{L}\_0 + \mu \mathcal{L}\_1 + \mu^2 \mathcal{L}\_2 + \dots,
$$

Here, zero-order approximation term, that is, L0 ¼ 0, would coincide exactly with the dispersion relation, accordingly to findings of Whitham [16]. The second term L1, entering this expansion, describes effects arising in the first-order nonlinear approximation analysis which we need to study here. Then, the nontrivial average Hamiltonian of the system would read

$$\mathcal{H} = \sum\_{k=1}^{3} \left( A\_{k,T} \frac{\partial \mathcal{L}\_1}{\partial A\_{k,T}} + \overline{A}\_{k,T} \frac{\partial \mathcal{L}\_1}{\partial \overline{A}\_{k,T}} \right) - \mathcal{L}\_{1,T}$$

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

$$\mathcal{H} = 2i\Omega \sum\_{j=1}^{2} w\_{j} \left(\overline{p\_{j}} - p\_{j}\right) \left|A\_{j}\right|^{2} + \epsilon n^{2} \left[\overline{A\_{1}}\overline{A\_{2}}A\_{3} \exp\left(i\Delta\omega T\right) + A\_{1}A\_{2}\overline{A\_{3}}\exp\left(-i\Delta\omega T\right)\right]. \tag{28}$$

Recall that the bending wave frequency ω ¼ ω1 ¼ ω2 should be close to the frequency ω3=2 that denotes the frequency of the axisymmetric radial oscillation, that is, ω3 ¼ 1. Note that the case under consideration is associated with the principal parametric resonance. His Hamiltonian produces a set of following evolution equations:

$$\frac{dA\_1}{dT} = \frac{2\Omega(\overline{p\_1} - p\_1)}{1 + p^2} A\_1 - \frac{i\epsilon \, n^2}{\omega(1 + p^2)} \overline{A\_2} A\_3 \mathbf{e}^{i\Delta\omega T}$$

$$\frac{dA\_2}{dT} = \frac{2\Omega(\overline{p\_2} - p\_2)}{1 + p^2} A\_1 - \frac{i\epsilon \, n^2}{\omega(1 + p^2)} \overline{A\_1} A\_3 \mathbf{e}^{i\Delta\omega T};\tag{29}$$

$$\frac{dA\_3}{dT} = -i\epsilon \, n^2 A\_1 A\_2 \mathbf{e}^{-i\Delta\omega T}$$

where pk ¼ �ð Þ1 k in 1 þ e2n2 � �= 1 þ e2 � �n2 � ω2 � � are the wave polarization coefficients which satisfy the equality p ¼ p1 � � � � ¼ p2 � � � � , due to the symmetry of the problem.

Now, we use the following transform of variables:

$$\begin{aligned} A\_1 &= a\_1 \exp\left(\frac{2\left(\overline{p\_1} - p\_1\right)}{1 + p^2} \bigg| \begin{matrix} \Omega(\varepsilon) d\varepsilon \\ 0 \end{matrix} \right); \\\ A\_2 &= a\_2 \exp\left(\frac{2\left(\overline{p\_2} - p\_2\right)}{1 + p^2} \bigg| \begin{matrix} \Omega(\varepsilon) d\varepsilon \\ 0 \end{matrix} \right); \\\ A\_3 &= a\_3. \end{aligned} \tag{30}$$

db1

db2

db3

dψ

dw3 dT ¼ �en2

dT ¼ �ηb1 þ en2b2b3 sinψ

dT ¼ �ηb2 þ en2b1b3 sinψ

dT ¼ �Δω þ �2n2e b2

motions; the first can be written as

while the second would be

ω 1 þ p2 ð Þ ;

ω 1 þ p2 ð Þ ;

dT ¼ �ηb3 � en2 b1b2 sinψ þ Q sin δT � w3 ð Þ=2;

b1 ¼ b2 ¼

Here, we have introduced a new notation: K ¼ Qn2

the problem of installation of the solid-state wave gyro.

vð Þ¼� τ; w 4pb1 cos ð Þ ψ τð Þ� nw cos ω þ μ

wð Þ¼� τ; w 4b1 sin ð Þ ψ τð Þ� nw cos ω þ μ

1 ω 1 þ p2 � �b2

b1b2 cosψ=b3 þ Q cos δT � w3 ð Þ=2b3,

ffiffiffi 2 p ηω ffiffiffiffiffiffiffiffiffiffiffiffiffi

number by analogy with Reynolds numbers in the hydrodynamics [18, 19].

2 � b2 3 2b2 3

where ψð Þ¼ T w3ð Þ� T w2ð Þ� T w1ð Þþ T ΔωT. These equations are convenient in the study of stationary damped forced motions performed in the system (32). There are two subsets of such

b1 ¼ b2 ¼ 0; b3 ¼ K 1 þ p2 � �ηω

1 þ p2 p ffiffiffiffiffiffiffiffiffiffiffiffi K � 1 p

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 0 ≤K ≤ K∗. Here, K∗ ¼ 1 denotes the critical value of the control parameter governing the bifurcations. This stationary solution coincides exactly with the solution if neglecting all the nonlinear terms in the system (19). Although near the point K∗ this stationary solution loses its stability to give its place for the new stationary state (35). The new stationary state would be stable at K > 1. It is not now surprising that as if the bifurcation parameter K grows even further, then the energy would be pumped only into the low-frequency bending modes while the amplitude of the high-frequency axisymmetric radial mode remains to be fixed, that is, b3 ¼ ηω 1 þ p2 � �= en2 � �. Probably, this mechanism represents an effective way in

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

δ þ Δω 2 � � � � τ þ O μ2

� � � � τ � 2b3 sin ð Þþ ϖτ O μ2

δ þ Δω 2

� � cosψ þ ωb1b2Q cos δT � w3 ð Þ 1 þ p2 � � 2ω 1 þ p2 ð Þb1b2b3

en2 ; b3 ¼ ηω 1 þ p2 � �

;

57

(33)

en2 , (34)

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

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e= 4 1 þ p2 � �η2ω2 � � for the characteristic

τ � �;

τ � �,

(36)

en2 : (35)

� � � b2

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

$$\frac{da\_1}{dT} = -\frac{i\epsilon n^2}{\omega(1+p^2)}\overline{a}\_2 a\_3 \mathbf{e}^{i\Delta\omega T};$$

$$\frac{da\_2}{dT} = -\frac{i\epsilon n^2}{\omega(1+p^2)}\overline{a}\_1 a\_3 \mathbf{e}^{i\Delta\omega T};\tag{31}$$

$$\frac{da\_3}{dT} = -i\epsilon n^2 a\_1 a\_2 \mathbf{e}^{-i\Delta\omega T},$$

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 Qv ¼ �2μηv\_; and Qw ¼ �2μ ηð Þ w\_ � Q cosϖτ into Eq. (19), we can obtain, after the exchange of variables (30), the following set of evolution equations:

$$\frac{da\_1}{dT} = -\eta a\_1 - \frac{i\epsilon \, n^2}{\omega(1+p^2)} \overline{a}\_2 a\_3 \mathbf{e}^{i\Delta\omega T};$$

$$\frac{da\_2}{dT} = -\eta a\_2 - \frac{i\epsilon \, n^2}{\omega(1+p^2)} \overline{a}\_1 a\_3 \mathbf{e}^{i\Delta\omega T};\tag{32}$$

$$\frac{da\_3}{dT} = -\eta a\_3 + i\mathbf{Q} \mathbf{e}^{i\delta T}/2 - i\epsilon \, n^2 a\_1 a\_2 \mathbf{e}^{-i\Delta\omega T}.$$

that can be rewritten, after one more exchange of variables; ajð Þ¼ T bjð Þ T eiwjð Þ T , in terms of realvalued amplitudes and phases:

$$\begin{aligned} \frac{db\_1}{dT} &= -\eta b\_1 + \frac{\epsilon n^2 b\_2 b\_3 \sin \psi}{\omega (1 + p^2)};\\ \frac{db\_2}{dT} &= -\eta b\_2 + \frac{\epsilon n^2 b\_1 b\_3 \sin \psi}{\omega (1 + p^2)};\\ \frac{db\_3}{dT} &= -\eta b\_3 - \epsilon n^2 b\_1 b\_2 \sin \psi + Q \sin \left(\delta T - \varphi\_3\right)/2;\\ \frac{d\psi}{dT} &= -\Delta \omega + \frac{-2n^2 \epsilon \left(b\_1^2 (\omega (1 + p^2) b\_2^2 - b\_3^2) - b\_2^2 b\_3^2\right) \cos \psi + \omega b\_1 b\_2 Q \cos \left(\delta T - \varphi\_3\right) \left(1 + p^2\right)}{2\omega (1 + p^2) b\_1 b\_2 b\_3};\\ \frac{d\varphi\_3}{dT} &= -\epsilon n^2 b\_1 b\_2 \cos \psi / b\_3 + Q \cos \left(\delta T - \varphi\_3\right) / 2b\_3. \end{aligned} \tag{33}$$

where ψð Þ¼ T w3ð Þ� T w2ð Þ� T w1ð Þþ T ΔωT. These equations are convenient in the study of stationary damped forced motions performed in the system (32). There are two subsets of such motions; the first can be written as

$$b\_1 = b\_2 = 0; \quad b\_3 = \frac{K(1+p^2)\eta\omega}{\epsilon n^2},\tag{34}$$

while the second would be

where pk ¼ �ð Þ1 k

satisfy the equality p ¼ p1

4.2. Resonant excitation of the gyro

valued amplitudes and phases:

of variables (30), the following set of evolution equations:

da1

da2

da3

� � � � ¼ p2 � � � �

Now, we use the following transform of variables:

A1 ¼ a1exp

56 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

A2 ¼ a2exp

A3 ¼ a3,

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

da1

da2

da3

in 1 þ e2n2 � �= 1 þ e2 � �n2 � ω2 � � are the wave polarization coefficients which

, due to the symmetry of the problem.

ð T

Ωð Þς dς

Ωð Þς dς

1 A;

1 A;

(30)

(31)

(32)

0

ð T

0

ω 1 þ p2 ð Þ a2a3eiΔωT;

ω 1 þ p2 ð Þ a1a3eiΔωT;

a1a2e�iΔωT,

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

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 Qv ¼ �2μηv\_; and Qw ¼ �2μ ηð Þ w\_ � Q cosϖτ into Eq. (19), we can obtain, after the exchange

ω 1 þ p2 ð Þ a2a3eiΔωT;

ω 1 þ p2 ð Þ a1a3eiΔωT;

a1a2e�iΔωT,

2 p1 � p1 � � 1 þ p2

2 p2 � p2 � � 1 þ p2

0 @

0 @

dT ¼ � ien2

dT ¼ � ien2

dT ¼ �ien2

becomes unstable triad with respect to small low-frequency perturbations [12].

dT ¼ �ηa1 � ien2

dT ¼ �ηa2 � ien2

dT ¼ �ηa3 þ iQeiδT=2 � ien2

that can be rewritten, after one more exchange of variables; ajð Þ¼ T bjð Þ T eiwjð Þ T , in terms of real-

$$b\_1 = b\_2 = \frac{\sqrt{2}\eta\omega\sqrt{1+p^2}\sqrt{K-1}}{\epsilon n^2}; \quad b\_3 = \frac{\eta\omega(1+p^2)}{\epsilon n^2}.\tag{35}$$

Here, we have introduced a new notation: K ¼ Qn2 e= 4 1 þ p2 � �η2ω2 � � for the characteristic number by analogy with Reynolds numbers in the hydrodynamics [18, 19].

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 0 ≤K ≤ K∗. Here, K∗ ¼ 1 denotes the critical value of the control parameter governing the bifurcations. This stationary solution coincides exactly with the solution if neglecting all the nonlinear terms in the system (19). Although near the point K∗ this stationary solution loses its stability to give its place for the new stationary state (35). The new stationary state would be stable at K > 1. It is not now surprising that as if the bifurcation parameter K grows even further, then the energy would be pumped only into the low-frequency bending modes while the amplitude of the high-frequency axisymmetric radial mode remains to be fixed, that is, b3 ¼ ηω 1 þ p2 � �= en2 � �. Probably, this mechanism represents an effective way in the problem of installation of the solid-state wave gyro.

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

$$\begin{aligned} \upsilon(\tau,\varphi) &= -4pb\_1 \cos\left(\psi(\tau) - n\varphi\right) \cos\left(\omega + \mu\left(\frac{\delta + \Delta\omega}{2}\right)\right)\tau + O(\mu^2 \tau); \\\ w(\tau,\varphi) &= -4b\_1 \sin\left(\psi(\tau) - n\varphi\right) \cos\left(\omega + \mu\left(\frac{\delta + \Delta\omega}{2}\right)\right)\tau - 2b\_3 \sin\left(\sigma\tau\right) + O(\mu^2 \tau), \end{aligned} \tag{36}$$

where

$$
\psi(\tau) = \frac{4p}{1+p^2} \int\_0^\tau \Omega(\varsigma) d\varsigma
$$

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

Bifurcation Makes a Wave Resonant Solid-State Gyro Be Stable

http://dx.doi.org/10.5772/intechopen.71726

59

\* and Dmitry Anatolyevich Kovriguine2

nonlinearity of the resonator in a post-bifurcation regime.

\*Address all correspondence to<EMAIL_ADDRESS>
Physica D: Nonlinear Phenomena. 1987;24(1–3):54-70

1 Radiophysics Faculty, Nizhny Novgorod State University n.a. N.I. Lobachevsky,

2 Applied Mathematics Chair, Nizhny Novgorod Technical State University n.a. R.E.

[1] https://www.grc.nasa.gov/www/K-12/airplane/dragsphere.html (Drag of a Sphere - NASA) [2] Lee J. Triad-angle locking in low-order models of the 2D Navier-Stokes equations.

[3] Sukhorukov AP. Nonlinear Wave Interactions in Optics and Radio Physics. Moscow:

Author details

References

Nauka; 1988

Svetlana Pavlovna Nikitenkova1

Alekseev, Nizhny Novgorod, Russia

Nizhny Novgorod, Russia

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

Figure 8. The wave precession. Modal number n ¼ 7. Solid lines refer to flexural mode, while the dashes to the circumferential one (η ¼ 0:1, μ ¼ e ≈ 0:01, κ ¼ 0:45, Ω ¼ 0:085, Q = 0.1). The dotted radius indicates the rotation angle Ωτ. The stiffness of the gyro platform κ is tuned to the seventh bending modes.

#### 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.

#### Author details

where

5. Conclusion

ψ τð Þ¼ 4p

58 Vibration Analysis and Control in Mechanical Structures and Wind Energy Conversion Systems

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

1 þ p2

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

Figure 8. The wave precession. Modal number n ¼ 7. Solid lines refer to flexural mode, while the dashes to the circumferential one (η ¼ 0:1, μ ¼ e ≈ 0:01, κ ¼ 0:45, Ω ¼ 0:085, Q = 0.1). The dotted radius indicates the rotation angle Ωτ. The