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doi: 10.3934/dcdsb.2020247

Analytical study of resonance regions for second kind commensurate fractional systems

Assia Boubidi , Sihem Kechida , and  Hicham Tebbikh

Laboratoire d'Automatique et Informatique de Guelma (LAIG), 8 Mai 1945-University, BP 401, 24000 Guelma, Algeria

* Corresponding author: Sihem Kechida

Received  May 2019 Revised  May 2020 Published  August 2020

The aim of this paper is to determine analytically the resonance limits for second kind commensurate fractional systems in terms of the pseudo damping factor $ \xi $ and the commensurate order $ v $ and in addition specify the different resonance regions. In the literature, these limits and regions have never been discussed mathematically, they are determined numerically. Second kind commensurate fractional systems are resonant if the equation : $ \Omega^{3v}+3\xi cos(v \pi/2)\Omega^{2v}+(2\xi^{2}+cos(v\pi))\Omega{^v}+\xi cos(v\pi/2) = 0 $, obtained by setting the first derivative of the amplitude-frequency response equal to zero, has at last one strictly positive root. As in the conventional case, resonance limits correspond to zero discriminant of the last equation. This discriminant is a cubic equation in $ \xi{^2} $ whose coefficients change depending on $ v $. To resolve this equation, the tangent trigonometric solving method is used and the relationship between $ \xi $ and $ v $ is established, which represents the resonance limits expression. To search resonance regions, a mathematical study is conducted on the first equation to find the positive roots number for each ($ v $, $ \xi $) combination. Compared to works already achieved, a new region appeared in the region of single resonant frequency with an anti-resonant one. The results are tested through numerical examples and applied to a fractional filter.

Keywords: Fractional order system, resonance, stability, cubic equation.
Mathematics Subject Classification: Primary: 11Y35; Secondary: 12E12.
Citation: Assia Boubidi, Sihem Kechida, Hicham Tebbikh. Analytical study of resonance regions for second kind commensurate fractional systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020247
References:
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M. Aoun, A. Aribi, S. Najar and M. N. Abdelkrim, On the fractional systems' fault detection: A comparison between fractional and rational residual sensitivity, in Eighth International Multi-Conference on Systems, Signals & Devices, Sousse, (2011), 1–6. doi: 10.1109/SSD.2011.5767424.  Google Scholar

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W. Du, Q. Miao, L. Tong and Y. Tang, Identification of fractional-order systems with unknown initial values and structure, Phys. Lett. A, 381 (2017), 1943-1949.  doi: 10.1016/j.physleta.2017.03.048.  Google Scholar

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M. R. Kumar, V. Deepak and S. Ghosh, Fractional-order controller design in frequency domain using an improved nonlinear adaptive seeker optimization algorithm, Comp. Sci., 25 (2017), 4299-4310.  doi: 10.3906/elk-1701-294.  Google Scholar

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R. Malti, M. Aoun, F. Levron and A. Oustaloup, Analytical computation of the $H_2$-norm of fractional commensurate transfer functions, Automatica J. IFAC, 47 (2011), 2425-2432.  doi: 10.1016/j.automatica.2011.08.021.  Google Scholar

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[19]

D. Matignon, Stability properties for generalized fractional differential systems, in ESAIM Proc., Soc. Math. Appl. Indust., Paris, (1998), 145–158. doi: 10.1051/proc:1998004.  Google Scholar

[20]

F. Merrikh-Bayat, M. Afshar and M. Karimi-Ghartemani, Extension of the root-locus method to a certain class of fractional-order systems, ISA. Trans., 48 (2009), 48-53.  doi: 10.1016/j.isatra.2008.08.001.  Google Scholar

[21]

C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-order Systems and control: Fundamentals and Applications, Advances in Industrial Control. Springer, London, 2010. doi: 10.1007/978-1-84996-335-0.  Google Scholar

[22]

A. Oustaloup, La Commande CRONE, Edition, Hermès, Paris, France, 1991. Google Scholar

[23]

I. Podlubny, Fractional-order systems and and $PI^\lambda D^\mu$-controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar

[24]

T. Poinot and J.-C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.  doi: 10.1007/s11071-004-3751-y.  Google Scholar

[25]

A. G. Redwan, A. S. Elwakil and A. M. Soliman, On the generalization of second-order filters to the the fractional-order domain, J. Circuit Syst. Comp., 18 (2009), 361-386.  doi: 10.1142/S0218126609005125.  Google Scholar

[26]

A. G. Redwan, A. M. Soliman and A. S. Elwakil, First-order filters generalized to the fractional domain, J. Circuit Syst. Comp., 17 (2008), 55-66.  doi: 10.1142/S0218126608004162.  Google Scholar

[27]

J. Sabatier, O. Cois and A. Oustaloup, Modal placement control method for fractional systems: Application to a testing bench, in IEEE 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, USA, (2003), 633–639. doi: 10.1115/DETC2003/VIB-48373.  Google Scholar

[28]

J. Sabatier, Ch. Farges and J.-C. Trigeassou, A stability test for non-commensurate fractional order systems, Syst. Control. Lett., 62 (2013), 739-746.  doi: 10.1016/j.sysconle.2013.04.008.  Google Scholar

[29]

P. Shah and S. Agashe, Review of fractional PID controller, Mechatronics, 38 (2016), 29-41.  doi: 10.1016/j.mechatronics.2016.06.005.  Google Scholar

[30]

J.-C. Trigeassou and N. Maamri, A new approach to the stability of linear fractional systems, in 2009 6th International Multi-Conference on Systems, Signals and Devices, Tunisia, (2009), 1–14. doi: 10.1109/SSD.2009.4956821.  Google Scholar

[31]

T. Yinggan, L. Haifang, W. Weiwei, L. Qiusheng and G. Xinping, Parameter identification of fractional order systems using block pulse functions, Signal Process., 107 (2015), 272-281.  doi: 10.1016/j.sigpro.2014.04.011.  Google Scholar

[32]

L. Zeng, P. Cheng and W. Yong, Subspace identification for commensurate fractional order systems using instrumental variables, in IEEE 2011 Control Conference, China, (2011), 1636–1640. doi: 10.1007/s12555-012-0511-5.  Google Scholar

[33]

W. Zhong, J. Lu and Y. Miao, Fault detection observer design for fractional-order systems, in 29th Chinese Control And Decision Conference (CCDC), China, (2017), 2796–2801. doi: 10.1109/CCDC.2017.7978988.  Google Scholar

[34]

S. Zhou, J. Cao and Y. Chen, Genetic algorithm-based identification of fractional-order systems, Entropy, 15 (2013), 1624-1642.  doi: 10.3390/e15051624.  Google Scholar

show all references

References:
[1]

M. Aoun, Systèmes Linéaires non Entiers et Identification par Bases Orthogonales non Entières, Ph.D thesis, Université Bordeaux 1, Talence, France, 2005. Google Scholar

[2]

M. Aoun, A. Aribi, S. Najar and M. N. Abdelkrim, On the fractional systems' fault detection: A comparison between fractional and rational residual sensitivity, in Eighth International Multi-Conference on Systems, Signals & Devices, Sousse, (2011), 1–6. doi: 10.1109/SSD.2011.5767424.  Google Scholar

[3]

A. Aribi, C. Farges, M. Aoun, P. Melchior, S. Najar and M. N. Abdelkrim, Fault detection based on fractional order models: Application to diagnosis of thermal systems, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 3679-3693.  doi: 10.1016/j.cnsns.2014.03.006.  Google Scholar

[4]

H. Atitallah, A. Aribi and M. Aoun, Diagnosis of time-delay fractional systems, in 17th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), Sousse, (2016), 284–292. doi: 10.1109/STA.2016.7952042.  Google Scholar

[5]

A. Ben Hmed, M. Amairi and M. Aoun, Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 842-865.  doi: 10.1016/j.cnsns.2014.07.014.  Google Scholar

[6]

J. Cao, J. Liang and B. Cao, Optimization of fractional order PID controllers based on genetic algorithms, in IEEE 2005 Machine Learning and Cybernetics, China, (2005), 5686–5689. doi: 10.1109/ICMLC.2005.1527950.  Google Scholar

[7]

D. R. Curtiss, Recent extensions of Descartes' rule of signs, Ann. of Math., 19 (1918), 251-278.  doi: 10.2307/1967494.  Google Scholar

[8]

W. Du, Q. Miao, L. Tong and Y. Tang, Identification of fractional-order systems with unknown initial values and structure, Phys. Lett. A, 381 (2017), 1943-1949.  doi: 10.1016/j.physleta.2017.03.048.  Google Scholar

[9]

A. S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area, IEEE Circuits and Systems Magazine, 10 (2010), 40-50.  doi: 10.1109/MCAS.2010.938637.  Google Scholar

[10]

T. T. Hartley and C. F. Lorenzo, A solution to the fundamental linear fractional order differential equation, Technical Report, 1998. Available from : https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990041952.pdf Google Scholar

[11]

M. Ikeda and S. Takahashi, Generalization of Routh's algorithm and stability criterion for non-integer integral system, Electron. Comm. Japan, 60 (1977), 41-50.  doi: 10.1155/2008/419046.  Google Scholar

[12]

E. Ivanova, X. Moreau and R. Malti, Stability and resonance conditions of second-order fractional systems, J. Vib. Control, 24 (2018), 659-672.  doi: 10.1177/1077546316654790.  Google Scholar

[13]

F. Khemane, Estimation Fréquentielle par Modèle Non-entier et Approche Ensembliste: Application à la Modélisation de la Dynamique du Conducteur, Ph.D thesis, Université Bordeaux 1, Talence, France, 2011. Google Scholar

[14]

M. R. Kumar, V. Deepak and S. Ghosh, Fractional-order controller design in frequency domain using an improved nonlinear adaptive seeker optimization algorithm, Comp. Sci., 25 (2017), 4299-4310.  doi: 10.3906/elk-1701-294.  Google Scholar

[15]

J. Lin, Modélisation et Identification des Systèmes d'ordre non Entier, Ph.D thesis, Université de Poitiers, Poitiers, France, 2001. Google Scholar

[16]

R. Malti, A note on ${L}_p$-norms of fractional systems, Automatica J. IFAC, 49 (2013), 2923-2927.  doi: 10.1016/j.automatica.2013.06.002.  Google Scholar

[17]

R. Malti, M. Aoun, F. Levron and A. Oustaloup, Analytical computation of the $H_2$-norm of fractional commensurate transfer functions, Automatica J. IFAC, 47 (2011), 2425-2432.  doi: 10.1016/j.automatica.2011.08.021.  Google Scholar

[18]

R. Malti, X. Moreau, F. Khemane and A. Oustaloup, Stability and resonance conditions of elementary fractional transfer functions, Automatica J. IFAC, 47 (2011), 2462-2467.  doi: 10.1016/j.automatica.2011.08.029.  Google Scholar

[19]

D. Matignon, Stability properties for generalized fractional differential systems, in ESAIM Proc., Soc. Math. Appl. Indust., Paris, (1998), 145–158. doi: 10.1051/proc:1998004.  Google Scholar

[20]

F. Merrikh-Bayat, M. Afshar and M. Karimi-Ghartemani, Extension of the root-locus method to a certain class of fractional-order systems, ISA. Trans., 48 (2009), 48-53.  doi: 10.1016/j.isatra.2008.08.001.  Google Scholar

[21]

C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-order Systems and control: Fundamentals and Applications, Advances in Industrial Control. Springer, London, 2010. doi: 10.1007/978-1-84996-335-0.  Google Scholar

[22]

A. Oustaloup, La Commande CRONE, Edition, Hermès, Paris, France, 1991. Google Scholar

[23]

I. Podlubny, Fractional-order systems and and $PI^\lambda D^\mu$-controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.  Google Scholar

[24]

T. Poinot and J.-C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.  doi: 10.1007/s11071-004-3751-y.  Google Scholar

[25]

A. G. Redwan, A. S. Elwakil and A. M. Soliman, On the generalization of second-order filters to the the fractional-order domain, J. Circuit Syst. Comp., 18 (2009), 361-386.  doi: 10.1142/S0218126609005125.  Google Scholar

[26]

A. G. Redwan, A. M. Soliman and A. S. Elwakil, First-order filters generalized to the fractional domain, J. Circuit Syst. Comp., 17 (2008), 55-66.  doi: 10.1142/S0218126608004162.  Google Scholar

[27]

J. Sabatier, O. Cois and A. Oustaloup, Modal placement control method for fractional systems: Application to a testing bench, in IEEE 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, USA, (2003), 633–639. doi: 10.1115/DETC2003/VIB-48373.  Google Scholar

[28]

J. Sabatier, Ch. Farges and J.-C. Trigeassou, A stability test for non-commensurate fractional order systems, Syst. Control. Lett., 62 (2013), 739-746.  doi: 10.1016/j.sysconle.2013.04.008.  Google Scholar

[29]

P. Shah and S. Agashe, Review of fractional PID controller, Mechatronics, 38 (2016), 29-41.  doi: 10.1016/j.mechatronics.2016.06.005.  Google Scholar

[30]

J.-C. Trigeassou and N. Maamri, A new approach to the stability of linear fractional systems, in 2009 6th International Multi-Conference on Systems, Signals and Devices, Tunisia, (2009), 1–14. doi: 10.1109/SSD.2009.4956821.  Google Scholar

[31]

T. Yinggan, L. Haifang, W. Weiwei, L. Qiusheng and G. Xinping, Parameter identification of fractional order systems using block pulse functions, Signal Process., 107 (2015), 272-281.  doi: 10.1016/j.sigpro.2014.04.011.  Google Scholar

[32]

L. Zeng, P. Cheng and W. Yong, Subspace identification for commensurate fractional order systems using instrumental variables, in IEEE 2011 Control Conference, China, (2011), 1636–1640. doi: 10.1007/s12555-012-0511-5.  Google Scholar

[33]

W. Zhong, J. Lu and Y. Miao, Fault detection observer design for fractional-order systems, in 29th Chinese Control And Decision Conference (CCDC), China, (2017), 2796–2801. doi: 10.1109/CCDC.2017.7978988.  Google Scholar

[34]

S. Zhou, J. Cao and Y. Chen, Genetic algorithm-based identification of fractional-order systems, Entropy, 15 (2013), 1624-1642.  doi: 10.3390/e15051624.  Google Scholar

Figure 1.  Stability and Resonance regions of the second kind commensurate fractional systems in the $ v $$ \xi $-plane [18]
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Figure 2.  Solution in $ y $ of equation $ D = 0 $
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Figure 3.  Division of $ v $$ \xi $-plane according to discriminant signs
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Figure 4.  Coefficients signs of $ f(x) $ and $ f(-x) $ in the $ v $$ \xi $-plane
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Figure 5.  Division of the $ v $$ \xi $-plane according to $ D $ signs and coefficients signs of $ f(x) $ and $ f(-x) $
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Figure 6.  Roots of (8) for different regions of the $ v $$ \xi $-plane according to Figure 3 division
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Figure 7.  Division of the $ v $$ \xi $-plane according to the number of strictly positive roots
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Figure 8.  Division of the $ v $$ \xi $-plane according to the number of strictly positive roots for stable systems
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Figure 9.  New division of resonance regions of the second kind commensurate fractional systems in the $ v $$ \xi $-plane
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Figure 10.  Magnitude Bode diagram for each region
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Figure 11.  Sallen-Key FLPF circuit
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Table 1.  Application of Descartes Rule of Signs according to Figure 5 division
Regions 1 2 31 32 41 42 5 6
$ D $ sign + + - - - - + +
$ R $ Nbr 3 3 1 1 1 1 3 3
$ f(x) $ coefs signs + + + + + - + - + + + + + + - + + - + - + - - - + + - + + - - -
Sign changes Nbr 0 3 0 2 3 1 2 1
$ R^{+} $ Nbr Possibility 0 3 or 1 0 2 or 0 3 or 1 1 2 or 0 1
$ f(-x) $ coefs signs - + - + - - - - - + - + - + + + - - - - - - + - - + + + - - + -
Sign changes Nbr 3 0 3 1 0 2 1 2
$ R^{-} $ Nbr Possibility 3 or 1 0 3 or 1 1 0 2 or 0 1 2 or 0
R+ Nbr 0 3 0 0 1 1 2 1
$ R^{-} $ Nbr 3 0 1 1 0 0 1 2
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Regions 1 2 31 32 41 42 5 6
$ D $ sign + + - - - - + +
$ R $ Nbr 3 3 1 1 1 1 3 3
$ f(x) $ coefs signs + + + + + - + - + + + + + + - + + - + - + - - - + + - + + - - -
Sign changes Nbr 0 3 0 2 3 1 2 1
$ R^{+} $ Nbr Possibility 0 3 or 1 0 2 or 0 3 or 1 1 2 or 0 1
$ f(-x) $ coefs signs - + - + - - - - - + - + - + + + - - - - - - + - - + + + - - + -
Sign changes Nbr 3 0 3 1 0 2 1 2
$ R^{-} $ Nbr Possibility 3 or 1 0 3 or 1 1 0 2 or 0 1 2 or 0
R+ Nbr 0 3 0 0 1 1 2 1
$ R^{-} $ Nbr 3 0 1 1 0 0 1 2
Table 2.  Positive roots number of (8) and corresponding regions
$ R^{+} $ Nbr 0 1 2 3
Regions 1 and 3 (31 and 32) 4 (41 and 42) and 6 5 2
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$ R^{+} $ Nbr 0 1 2 3
Regions 1 and 3 (31 and 32) 4 (41 and 42) and 6 5 2
Table 3.  Resonance regions of the second kind commensurate fractional systems
Region and defined domain Exemple $ (\omega_{n}=1) $
$ v=0.5 \;and\;\xi $=0.5
Gray region (8)
No resonant frequency $ \Downarrow $
$ \Omega^{1.5}+1.06\Omega^{1}+0.5\Omega^{0.5}+0.35=0 $
$\left\{\begin{array}{l}0<v \leq 0.5 {\;and\;} 0 \leq \xi \\ 0.5<v \leq 1 {\;and\;} \xi_{r l} \leq \xi\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=-0.87 \\ \Omega_{2}=-0.37+i 0.08 \\ \Omega_{3}=-0.37-i 0.08\end{array}\right.$
Yellow region $ v $=1.5 and $ \xi $=0.9
A single resonant frequency (8)
$ \Downarrow $
$ \Omega^{4.5}-1.91\Omega^{3}+1.62\Omega^{1.5}-0.64=0 $
$\left\{\begin{array}{l}0<v<1 {\;and\;} cos (v \pi / 2)<\xi<0 \\ 0.5<v<1 {\;and\;} \xi=0 \\ v=1 {\;and\;} 0<\xi<0.7071 \\ 1<v \leq 1.7837 {\;and\;}-cos (v \pi / 2)<\xi \\ 1.7837<v<2 {\;and\;}-cos (v \pi / 2)<\xi \leq \xi_{r l}\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.93 \\ \Omega_{2}=0.73+i 0.52 \\ \Omega_{3}=0.73-i 0.52\end{array}\right.$
$ \Rightarrow \omega_{res} =0.93 rd/sec $
$ v=0.8 \;and \;\xi=0.28 $
Brown region (8)
A single resonant frequency with anti-resonant frequency $ \Downarrow $
$ \Omega^{2.4}+0.26\Omega^{1.6}-0.65\Omega^{0.8}+0.09=0 $
$0.5<v<1$ and $0<\xi<\xi_{r l}$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.09 \\ \Omega_{2}=0.52 \\ \Omega_{3}=-1.00\end{array}\right.$
$\Rightarrow\left\{\begin{array}{l}\omega_{{anti}-r e s}=0.09 {rd} / {sec} \\ \omega_{ {res}}=0.52 {rd} / {sec}\end{array}\right.$
$ v=1.9 \;and \;\xi=2.5 $
Green region (8)
Two resonant frequencies $ \Downarrow $
$ \Omega^{5.7}-7.41\Omega^{3.8}+13.45\Omega^{1.9}-2.47=0 $
$1.7837<v<2$ and $\xi_{r l}<\xi$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.44 \\ \Omega_{2}=1.65 \\ \Omega_{3}=2.23\end{array}\right.$
$\Rightarrow\left\{\begin{array}{l}\omega_{r e s 1}=0.44 r d / s e c \\ \omega_{\text {anti}-r e s}=1.65 r d / s e c \\ \omega_{\text {res} 2}=2.23 r d / \text {sec}\end{array}\right.$
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Region and defined domain Exemple $ (\omega_{n}=1) $
$ v=0.5 \;and\;\xi $=0.5
Gray region (8)
No resonant frequency $ \Downarrow $
$ \Omega^{1.5}+1.06\Omega^{1}+0.5\Omega^{0.5}+0.35=0 $
$\left\{\begin{array}{l}0<v \leq 0.5 {\;and\;} 0 \leq \xi \\ 0.5<v \leq 1 {\;and\;} \xi_{r l} \leq \xi\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=-0.87 \\ \Omega_{2}=-0.37+i 0.08 \\ \Omega_{3}=-0.37-i 0.08\end{array}\right.$
Yellow region $ v $=1.5 and $ \xi $=0.9
A single resonant frequency (8)
$ \Downarrow $
$ \Omega^{4.5}-1.91\Omega^{3}+1.62\Omega^{1.5}-0.64=0 $
$\left\{\begin{array}{l}0<v<1 {\;and\;} cos (v \pi / 2)<\xi<0 \\ 0.5<v<1 {\;and\;} \xi=0 \\ v=1 {\;and\;} 0<\xi<0.7071 \\ 1<v \leq 1.7837 {\;and\;}-cos (v \pi / 2)<\xi \\ 1.7837<v<2 {\;and\;}-cos (v \pi / 2)<\xi \leq \xi_{r l}\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.93 \\ \Omega_{2}=0.73+i 0.52 \\ \Omega_{3}=0.73-i 0.52\end{array}\right.$
$ \Rightarrow \omega_{res} =0.93 rd/sec $
$ v=0.8 \;and \;\xi=0.28 $
Brown region (8)
A single resonant frequency with anti-resonant frequency $ \Downarrow $
$ \Omega^{2.4}+0.26\Omega^{1.6}-0.65\Omega^{0.8}+0.09=0 $
$0.5<v<1$ and $0<\xi<\xi_{r l}$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.09 \\ \Omega_{2}=0.52 \\ \Omega_{3}=-1.00\end{array}\right.$
$\Rightarrow\left\{\begin{array}{l}\omega_{{anti}-r e s}=0.09 {rd} / {sec} \\ \omega_{ {res}}=0.52 {rd} / {sec}\end{array}\right.$
$ v=1.9 \;and \;\xi=2.5 $
Green region (8)
Two resonant frequencies $ \Downarrow $
$ \Omega^{5.7}-7.41\Omega^{3.8}+13.45\Omega^{1.9}-2.47=0 $
$1.7837<v<2$ and $\xi_{r l}<\xi$ $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.44 \\ \Omega_{2}=1.65 \\ \Omega_{3}=2.23\end{array}\right.$
$\Rightarrow\left\{\begin{array}{l}\omega_{r e s 1}=0.44 r d / s e c \\ \omega_{\text {anti}-r e s}=1.65 r d / s e c \\ \omega_{\text {res} 2}=2.23 r d / \text {sec}\end{array}\right.$
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